Simply because of the nature of the series, this arguably happens a lot in Axis Powers Hetalia. Examples may include the Nordics, micronations or characters that only have a basic design.
This is also arguably how the Allies and the more in-the-background-initially nations like Poland, Lithuania and the Frying Pangle (Austria, Hungary, Prussia) started off in the first place.
Yuka Hanazawa of Beelzebub was originally just one of two occasional characters in The Red Tails, usually just appearing for a quick background appearance. As of late, she's taken on a more prominent role, frequently seen hanging around Kanzaki.
In Black Cat, Mason Ordrosso goes from having a one-page cameo (in the manga), to becoming the second Big Bad in the anime adaptation.
Konori Mii from A Certain Scientific Railgun was originally a nameless Judgement character in the manga, but in the anime, she got backstory, character development, her own fair share of moments and voicing from Kana Ueda.
Uiharu is a very small character in Index but a main cast member in Railgun.
Nina Einstein is arguably a rather uncommon case on how this isn't always a good thing. At first she seemed to be nothing more than a quiet, shy little Meganekko whom is just one of Milly's friends. However, due to her knowledge in science (among other things), her role gradually gets considerably larger. But the bad news is that this is when her reputation of being "an evil insane racist" was starting to grow as well. (Granted, the show tries to redeem her later on in R2, but lets just say not everyone was willing to forgive her.) Though in a way people probably would've liked her more right from the start if she was just one of Milly's little friends and pretty much stayed that way (in fact, the manga adaptation actually does just this.)
In a way Lelouch and Nunally's mother Marianne is like this as well; in the beginning she seemed to be a token "Kind Mother whose only purpose is to have died in a flashback just so main characters can have a sad backstory". However later on in R2 let's just say that wasn't quite the case, and it appears that she is just as bad as her dear hubby Charles. Granted it was all a part of their rather convoluted plan for peace that reminded people about a certain other plan lets just say. Not to mention on how the only reason why Lelouch and Nunally were living their lives like they were before the whole story started is that it was all a part of Charles and Marianne's plan. So in short as for later on R2 Marianne went from one of the better mothers in the show to quite possibly the worst mother ever.
While some characters were Demoted to Extra's for "Lelouch of The Rebellion" (manga), some such as Euphemia were given far bigger parts.
Jeremiah Gottwald also got an expanded role in R2 after some Executive Meddling. Apparently, Jeremiah gained such a fanbase in the first season, that the producer rewrote the script to include him in R2 instead keeping him killed off. Itwasawesome.
The author of D.Gray-Man revealed in her commentary that Miranda Lotto was originally supposed to be an arc character never to be seen again, but as she got into the story she took a liking to her, made her an exorcist and had her join the main cast a couple of arcs later.
Takagi was originally an unnamed sidekick under Megure in the anime and Canon Immigrant; later show in the manga and ascended to major character status with his long love story with Satou.
Conan's wonderful toys tend to have a lot more prominence in the anime, especially the movies (in the manga, the only ones he uses regularly are the watch and the bow-tie). For the Most Triumphant Example: the turbo-powered skateboard that appears in just about every movie and quite a few of the anime-only Filler episodes? It appeared in a grand total of two storylines in the manga, which were more than seven hundred chapters apart (it was actually smashed to bits at the end of its debut arc, and for a long time it was implied that Dr. Agasa didn't bother fixing it).
Digimon Tamers had a few of these. Bridge Bunny Riley/Reika became an Ascended Extra later on in the series, although that might have been because she was dating the not-bad-guy of the series (Yamaki). Takato's friends Jeri, Kazu, and Kenta; as well as Henry's little sister Suzie, also become this trope when all four of them become Sixth Rangers.
Chi-Chi first appeared as the Ox King's sweet little daughter, and was little more then a running gag; she wanted to marry Goku, while Goku failed to understand that marriage wasn't a type of food. Years later, an angry and considerably skilled Chi-Chi fought Goku in the 23rd World Martial Arts Tournament. They end up together as husband and wife, where from Dragon Ball Z onwards, Chi-Chi goes from Action Girl to Mama Bear, while still getting to kick someone's ass occasionally.
Likewise, Bardock only appeared in one panel in the original manga, only to get his own TV special in the anime, and appear prominently in flashbacks as well.
The "World Tournament" Announcer. Considering he is always featured when a (Earthly and official) tournament comes up. He even ages alongside the other characters with each appearance.
In the Excel Saga manga, Pedro appears only a few panels before dying. He still dies as quickly in the anime, but his story continues, with continuity that the rest of the series lacks.
Shin, Kenshiro's first notable adversary in Fist of the North Star, was killed off rather quickly in the original manga after playing out his role to serve as exposition on Kenshiro's background and motivation, since the future of the serial was still uncertain at the time. In the TV series, Shin's story arc was extended beyond the events of the original manga, with Shin getting much greater development. One episode gives his own moment of glory when he singlehandedly thwarts a rebellion within his own subordinates, displaying a bit of his martial art skills before confronting Kenshiro.
Lust's role in the anime was greatly expanded, and given much more depth than her manga counterpart. In the anime she wants to be human because she feels something is missing, and she is one of the three Homonculi portrayed with great sympathy (Greed and Wrath are the others).
In the manga, Basque Grand is only mentioned posthumously and seen during the Ishbal flashback arc. He gets more exposure (and a personality change) here, and is in fact one of the main recurring antagonists early in the series.
Shou Tucker, a minor villain who was wasted in fairly short order in the manga, becomes a Chimera in the first anime and returns to cause trouble a few more times.
Rose, a minor Audience Surrogate from the beginning of the manga, became important to the story near the end.
The Chimeras from Devil's Nest also had greatly expanded roles in the anime, especially Marta. Marta, one of Greed's companions, who had a somewhat significant role in the latter half of the anime, even traveling with the Elrics for a few episodes, unlike the manga where she is killed with the rest of Greed's allies in the Devil's Nest.
In the manga, Sheska is barely in it after her introduction, with the exception of a few cameos. Sheska got a rather significant amount of screen time and basically became a second heroine next to her best friend Winry.
In the third season of Full Metal Panic!, the assassin twin sisters Yu Fan and Yu Lan had a much expanded role than they had in the novel on which the show was based. In said novel, they were male and died rather quickly. Perhaps their tendency to hold each other naked had something to do with this change.
In Gaiking: Legend of Daiku Maryu, Kain starts out as one of Nouza's subordinates, and is the only one to survive. During the final battle, he even pilots Vulking as a decoy to allow Nouza to sneak into the enemy base. In addition, Daiya's friend Naoto had only featured in the first episode, but makes a reappearance later on, piloting the Daiku Maryu after its regular helmsman is seriously injured.
Yuki Ushimaru from Gakuen Babysitters slowly rises from her status as a walking Running Gag when she gets a rival for Ryuuichi's affections, resulting in slightly more frequent and significant appearances afterwards.
There's also Kaoruko "Gyanko" Suzaku, who only had a brief appearance at the very end of Build Fighters admiring her older brother's newest Gyan Gunpla, only to become a major character in the next season. Time will only tell if her younger brother will show up next series.
Sae's sister Chika is arguably important in the Hidamari Sketch anime, but you won't be able to find her in the manga — she is only referenced to and not given a name there.
Sumomo in Sgt. Frog was, in the manga, a character from a one-shot side story loosely connected to the main comic. In the anime, she was remade into an alien Idol Singer who was taking an impromptu vacation on Earth in her first appearance, and eventually became a recurring character.
Interestingly, the show seems aware of this, as some characters only appear between long stretches of episodes, prompting the narrator to cheerfully jog the viewer's memory.
In the manga adaptation for Kingdom Hearts II, Sark was given more scenes and played a major role during the second half of the "Space Paranoids arc", including taking Tron captive and torturing him, and later ambushing the party when the DTD is accessed. Following his defeat, we learn that he once believed in the users like Tron does, but lost faith when it appeared that Ansem betrayed them, and warns Tron to let go of his faith while he still can, which causes Sora to feel sorry for Sark.
Renko Kamishakujii and Koyuki Asagiri, while they both had minor roles in the first Kujibiki Unbalance "series" (it's tough to tell with only three episodes), they were major characters in the TV series version. In contrast, Komaki and Izumi, once major characters, were reduced to side roles.
Mahou Sensei Negima! exemplifies this trope. Several characters who barely got any panels per volume to their selves for the first few volumes get entire sub-plots revolving around them. A few examples:
Natsumi, Kazumi, Ku Fei and Mana are barely mentioned in the beginning yet became pivotal players in the most recent story arc.
The defeat of Rakan revolved partly around Ako trying to work out her feelings for Negi/Nagi.
Chao Lingshen had an entire story arc written around her. If that's not ascension of an extra, I don't know what is.
Chie Harada was simply one of Those Two Guys in Mai-HiME, and a close friend who provided emotional and moral support to the protagonist, but didn't have much plot importance. Her Mai-Otome counterpart, Chie Hallard, is a high-profile supporting character, even getting pictured on a DVD cover with Aoi, That Other One Girl.
Not to mention Arika and Nina, the two main characters of Mai-Otome were practically nonexistent in Mai-HiME. They had no lines and minimal screen time - Arika only as a cameo in the finale, Nina as simply a Living Prop in Mai's classroom. Only Nina had a name, and it was in Japanese, only revealed in one of the Omake that briefly lists the names of everyone in Mai's class.
Madame Taki in Mermaid Melody Pichi Pichi Pitch. In the manga, she's a nondescript fortune teller from Lucia's palace who only appears twice, and one of those times was in a flashback detailing why Lucia was raised as a Sleeping Beauty. In the anime, she's a weird fortune teller who came to the surface with Lucia, Nikora and Hippo, and she's also Nikora's "guardian creature", like Hippo to Lucia.
In one of the Naruto video games, the ANBU organization, specifically members Towa, Komachi and Yugao Uzuki, were given major roles in the storyline. Also, Kakashi wearing an ANBU uniform is playable in the Naruto: Ultimate Ninja games.
Matsuri, formerly a nameless Fangirl of Gaara's in Part II, appears in the last of the Part 1 filler arcs as the first student to accept him as her teacher, and who gets kidnapped by the villains of the arc to lure him out. She also gets some lines of dialogue in the anime version of the "Kazekage Rescue Mission" and "Fourth Shinobi World War" arcs.
And proceeded to get her very own entry into the official databook under that name.
Chiriku, one of Asuma's old friends and one of the 12 Guardian Ninjas, gets more of a role in the filler "Temple of Fire" arc.
Rock Lee, despite his relatively sparse screentime post-Time Skip, did get his own spinoff in the form of Rock Lee's Springtime of Youth, which also prominently featured his teammates Neji and Tenten.
Lee's sensei Might Guy, who doesn't get to do much for most of the manga, ends up becoming pretty prominent during the final set of arcs; he single-handedly takes down one of the more dangerous villains left, and gets a decent amount of spotlight in the battle against the Big Bads (in fact, Guy almost single-handedly beats the crap out of Madara despite the latter's Story Breaker Powers). He even gets his own flashback chapter.
Hinata. Not including the fair amount of filler that feature her prominently, she gradually became more relevant in the actual manga, particular with regards to her appearances in the "Pain's Assault" and "Fourth Shinobi World War" arcs. That said, she was still relegated to the background with most of the other cast before the final arc, as the main four characters all took center stage once again. That said, The Last, the Grand Finale movie to the original manga, elevated her status to Naruto's main love interest, culminating in the two becoming an Official Couple.
Kaworu from Neon Genesis Evangelion gets this treatment in the Rebuild of Evangelion films, where he has a greater presence throughout the actual storyline. It sometimes takes fans a bit of time to realize that outside of the movies, he only appeared in one episode, and had the approximate screentime of Jet Alone (remember JA? We didn't think so.) That said, he was important to the plot, but most of the character attributes were retconned since they couldn't actually be displayed in what little screentime he had. He also gets an expanded role in the manga, though his characterization is very different.
Michael and Hoichael, two delinquents in Water 7, are once suspected as being responsible for the assassination attempt on Water 7, and later try to mug Zoro, only to get beaten up and forced to lead him back to the rest of the crew. In the Post-Enies Lobby arc, they appear in a filler episode as part of a makeshift family consisting of a woman and her "children," who are not blood-related, and Zoro helps defeat the collectors after they hire Mikazaki of the Crescent Moon (another Ascended Extra, a pirate with a 38 million bounty who got Worfed by the Galley-La company).
Silvers Rayleigh first appeared in a single panel at the beginning of the series. Fast forward almost 500 chapters later, and not only is he revealed to be Gold Roger's first mate, but he also helps train Luffy during the timeskip.
Boa Hancock. She's the second character after Luffy from One Piece who is included to the J Stars Victory VS character roster. What makes it really impressive is the fact that not even the other Straw Hats- yes, not even Zoro- are in the game. And not only that, Hancock is the second fightingfemale character despite not even a main character like Rukia or Kagura or Lala who are just supporting characters in the game.
A more recent example is Trafalgar Law, who appeared during the Sabaody Archipelago Arc as a minor character. He later shows up at the end of the Marineford War to aid an out-of-commission Luffy, then after a few arcs, reappears again as a major ally in the Punk Hazard Arc and Dressrosa Arc. Granted, it was hinted that the Eleven Supernovas would have bigger roles to play in future story events, but he practically becomes the Deuteragonist for a while.
Ouran High School Host Club's Renge Houshakuji was introduced early on with her own chapter, and then she faded to the background. She does still make occasional cameo appearances, though. Bisco Hatori had mentioned Renge was intended to be a recurring character, but for whatever reason it never happened. Her role in the anime is expanded quite a bit as an As You KnowFangirl, and she does a few of the tasks Tamaki had initially done in the manga (such as helping Nekozawa with his photophobia). Hatori was happy to see Renge given more screentime in the anime, even if her appearances became less frequent toward the finale.
Pandora Hearts: Poor, poor Elliot and Leo get this in the worst possible way with the former finally seeing the Tomato in the Mirror and realizing that he is the one who has been offing his family members all along, a revelation that leads to his Heroic Sacrifice, and the latter blaming himself for his best friend's death and eventually pulling a Face–Heel Turn because he can't handle the guilt, joining the Baskervilles and reawakening Glen Baskerville's soul inside him, thus becoming an antagonist to Oz, who had grown to see him as a friend.
In Best Wishes, they repeat this with Cilan and Iris.
Pikachu was nothing in Pokémon Red and Blue. As explained here, it was turned from an insignificant unevolved Pokémon into the poster boy for the entire thing.
Pokémon Special has this with nearly every Gym Leader, and quite a few other people. A noticeable example is Courtney, a Team Magma admin. She never appeared in the anime, never appeared in any other manga, doesn't have any official art, was fought once, and never appeared in the timeline canon Emerald. But she's the most developed of the admins for either team, maybe even more-so then the leaders.
For an "in house" example there's Professor Yanase who initially was just a scientist on the control Rayquaza project and later is revealed to be Yanase Berlitz, Platinum's mother.
Pokemon adaptations in general tend to do this. Any important NPC will probably have some plot significance for at least a chapter.
Sentaro Daimonji in Ranma ˝ was a minor character who appeared in a brief three-chapter-storyline early on in the manga. In the anime, he was made into a recurring character that generally popped up Once a Season.
General Gunther Reindardt of Robotech: Originally, the animation for the final series made many fans assume that the returning Expeditionary fleet was destroyed by the Invids' transcendence to a higher plane. The Jack McKinney novels assumed this, and it was probably true in the original Japanese version of Genesis Climber MOSPEADA. At that time, the General didn't yet have a name and his connection to Admiral Hunter was Just Following Orders. He was named in the novels and was established as one of Rick Hunter's close friends. After 20 years of debating what was seen in the animation, the 2007 movie Robotech: The Shadow Chronicles not only made Reinhardt one of the lead characters, but showed that the Invids' transcendence destroyed only the Neutron-S missiles, and not any manned Expeditionary ships, ensuring that Reinhardt and his crew survive to be in sequels.
The entire Montague family in the Romeo X Juliet anime. In the original play, the Capulets had more focus, partially due to Juliet's Arranged Marriage to Paris; in contrast, Romeo's mother, Lady Montague, has one spoken line and dies from sadness by the end of the play. In the anime, it's reversed: save Juliet, all of the Capulets are dead, Lord Montague is the Big Bad, Romeo's the one with an arranged suitor, and Tybalt is secretly related to Romeo as a cousin, whereas he was Juliet's in the play. For individual characters, there's Romeo's relative Benvolio, and Juliet's nurse, who is actually given a name, Cornelia.
The fourth season has an especially odd example. The Amazon Trio (three goofy bishounen) stuck around long enough that fans not familiar with the manga might have thought they were the season's primary Quirky Miniboss Squad, even though they would be replaced by the more appropriate Amazoness Quartet (who actually got less appearances.) In the manga, the members of the Trio were one-shot characters who served the Amazoness Quartet and were generally closer to the Monster of the Week type.
As hinted at above, the Amazoness Quartet are an inversion — not only are they the main secondary villains of that Story Arc in the manga, but they're some of the very few villains in the manga to be redeemed, and end up becoming the Asteroid Senshi who act as Chibiusa's bodyguards. In the anime, they have less episodes than the Trio did, and never show up again after their redemption, thus never becoming the Asteroid Senshi. Thus, the Quartet were actually Demoted to Extra from manga to anime.
Another noticeable example is the first season's Nephrite, whose goal was to locate the Silver Crystal. While he was quickly and unceremoniously killed off in the manga, in the anime, Usagi/Serena's friend Naru/Molly kept interfering in his progress, and through continued interaction, he fell for her, and pulled a last-minute Heel–Face Turn to save her at the cost of his own life.
The Sailor Starlights were even noted for being this trope by the author of the original manga (where they're killed off after a few chapters, never to be seen again). Their anime role is basically a mix of Tuxedo Mask and the Outer Senshi, with Seiya's relationship with Usagi given a much bigger focus (due to Seiya being rewritten into a boy who transforms into a girl, rather than a girl crossdressing as a boy.) Oh, and they're the only ones apart from Usagi to survive until the very end.
Sailor Moon Crystal did this again with the Four Generals. Despite the new anime being heavily advertised as being true to the original manga's events without filler, the Four Generals were once again given bigger roles. While most of them died after only two or three chapters of being there, they now remained alive until the end of the Dark Kingdom arc and being officially labelled as having been in love with Serenity's guardians Mercury, Mars, Jupiter and Venus.
This is not only ascending their roles again, but actually going againstTakeuchi Naoko's stance that, while she did once draw a picture where each of the girls were paired up with one of the Four Generals, she never intended or put any romance between them into the manga... except in Sailor V and hinted at in the Sailor Moon manga, where Venus had a crush on Kunzite, though whether this was reciprocated is unknown.
In the School-Live! Taroumaru was a puppy who died in the chapter he appeared in. In the anime he is the Team Pet.
When Slayers was novelized, Lina and Gourry were actually the only real protagonists; their companions in the first arc of novels, Zelgadis and Amelia, were only their allies for that period of time (and even then, Amelia did not appear during the battle with Copy Rezo as she did in the anime). They were replaced with Luke and Milina in the second arc of novels. When the anime was becoming popular, however, Zel and Amelia became popular enough to appear in more places than they did, and upon expansion, the "Slayers" became a four-man party instead; just about all of the video games and manga that came followed suit with this. By contrast, Luke and Milina only appeared outside the novels in the Hourglass of Falces manga, and even there, the group is a Six-Man Band instead.
Shiro Sanada in Space Battleship Yamato: Originally, he was just the ships Smart Guy. And due to the fact that he was a little older than the mostly young crew, it was expected that he would die as is common for mature characters in an ensemble cast of mostly youngsters (so that the young characters can grow and develop). Sanada shares with Kodai the distinction of being the Yamato Sagas two most longeval characters. He reunites with Kodai in the 2010 Yamato Ressurection in which Sanada now outranks captian Kodai. His character is even more developed in Space Battleship Yamato 2199 where he is now the Captain's true first officer as is appropriate for his age and experience. He also has an ongoing romantic interest, something that was totally ignored in the original series.
Kazuhiko Amagasaki from Tenchi Muyo! only showed up in a brief scene of episode 1, but managed to gain enough popularity to get a bigger role in Tenchi in Tokyo.
Ignacio Axis in the animated adaptation of Toaru Hikuushi e no Koiuta. The anime seems to be treating him as one of four protagonists, if promotional material is anything to go by. In the novels, however, Ignacio is considered a supporting character only, with the original story focusing on Kal-el, Claire and Ariel.
Tokyo Ghoul uses this as a particular stunning twist in the sequel, :Re. Remember Seidou Takizawa, the Plucky Comic Relief from the CCG that was brutally killed off during the finale? Turns out he was Not Quite Dead, and ended up Reforged into a Minion by Aogiri. He quickly establishes himself as an Evil Counterpart to Sasaki, and the main villain of the arc with nearly 11 chapters of carnage to his credit. Since he got away, it's clearly not the last we'll see of him. Unlike most examples, subtle Foreshadowing hints that this twist was intended from the very beginning.
Following a certain point in the first manga, Yui was given increasingly greater exposure even as Haruna (principal Love Interest of the male protagonist) appeared less and less.
Kyoko was born as fictonial character of her Show Within a Show (a parody of the Magical Girl). After, she appears as real character and became Run's best friend. In "Darkness" she is also promoted to new potential harem member of Rito.
"Lum's Stormtroopers" in Urusei Yatsura. In the manga, they were minor characters who didn't even have nicknames given, weren't really distinguished from each other (to the point that it's not consistent about whether there are four of five of them), and stop appearing fairly early on in the series. In the anime, they're given the nicknames "Megane", "Chibi", "Kakugari", and "Perm", Megane is a supporting character about as important as (for example) Mendo or Ten, and Perm is sort of a composite of two characters from the manga: minor character Kosuke Shirai, and the nameless permed-hair guy. (Kosuke Shirai himself eventually appeared in the anime in the 2008 OVA.)
Despite being one of the lesser-known characters of the franchise, Armor was a main character in Marvel Anime: X-Men. This lead to her getting some more spotlight in the comics, such as her major role in the Alpha and Omega mini-series.
Miho Nosaka only appeared in one chapter of the manga, but was made a main character in the Toei anime version (the lost "season zero"). Her personality was drastically altered, too; in the manga, she was a gentle, bookish girl, but in the anime she is sweet, kind, innocent, outgoing, and a Kawaiiko.
Gozaburo Kaiba only appeared briefly in a flashback in the manga. In the anime, not only were the flashbacks expanded, he was the Big Bad of the Virtual Nightmare Arc. (Which was not in the manga.)
Ryuji Otogi (Duke Devlin in the English version). In the manga, his father sent him to defeat Yugi because his loss to Yugi's grandfather Sugoroku when they played the Devil's Board Game, which aged him 50 years overnight. But Otogi isn't that cold-hearted and won't let his dad make him hate Yugi. As for the anime, he was first shown wanting revenge on Yugi for defeating Pegasus at Duelist Kingdom. After being beaten at his own game, Dungeon Dice Monsters, he softens up completely and appears throughout the rest of the anime as a supporting character.
Remember Tetsu Ushio, that bully who was Dark Yugi's first victim? He's back as a policeman in a supporting role in 5Ds, and even gets to act heroic.
Kaiba himself becomes one in the anime version of the Memory World Arc. In the manga after Battle City he's Put on a Bus (or in his case, a jet) and only makes a brief cameo in the very last chapter. On the other hand, in the anime he'd already been upgraded from a major supporting character as he was in the manga into the deuteragonist on the show next to Yugi (or tritagonist if you count Yugi and Yami as two characters) and was already greatly involved in just about every filler arc in the series, so the writers probably felt it would be weird if he missed out on the finale. Thus, not only does he wind up dueling Bakura at the beginning of the storyline but eventually he winds up following Yugi and the others into the past as well, ultimately helping Atem take on Zorc and getting a Big Damn Heroes moment by the end.
In Yu-Gi-Oh!: Capsule Monsters, while they're always present in the main series, Anzu and Honda actually get to contribute to events. Sugoroku is a straight example.
Insector Haga and Dinosaur Ryuzaki (Weevil Underwood and Rex Raptor in the English version) were minor duelists in the original manga; Haga lost to Yugi and Jonouchi respectively during the two tournament arcs while Ryuzaki lost to Jonouchi in the first and was defeated off-panel in the second. The anime meanwhile upgraded their status to Those Two Bad Guys and made them recurring rivals to Yugi and Jonouchi respectively long after the point they'd been written out of the manga, appearing all the way up to the anime's final episode as part of its "Where Are They Now?" Epilogue during the credits.