The excellent Fan Remake of King's Quest II does this with several NPCs. Perhaps the best is Valanice, who was rewritten from her original flat, MacGuffin-esque Distressed Damsel portrayal to a character more in line with what's seen in her later appearances in the series, who — although a prisoner — is hardly a helpless shrinking violet. Similarily, the vampire (originally Dracula; Caldaur in the remake) is turned into a sympathic character with personality, depth and motives, who will end up aiding the hero if you help him in return. Seeing as his original role was to hang out in his castle until the hero showed up to kill him for no apparent reason (other than needing the key he had), this was a definite improvement.
In the original versions of the series, Rosella was given her own adventure to star in after appearing briefly in the previous game as the Distressed Damsel, and later on she co-starred alongside Valanice as the heroes of King's Quest VII. Princess Cassima, meanwhile, was introduced near the end of King's Quest V and was later ascended to being a major character in King's Quest VI.
Waluigi is a mild example. Originally showing up in Mario Tennis as a fill in for Wario's partner and Luigi's rival, Waluigi's shown up in virtually every spinoff since then, and is the only regular character that hasn't ever been in a platformer to do so. He's either loved for being insane or loathed as a auxiliary character.
World of Warcraft does this constantly. Literally every pre-WoW bit of Warcraft material had a lot of lore that may or may not have appeared on screen, and may have had little if any effect on the plot, that end up becoming major characters in a later game or when Blizzard needs material for an expansion pack or raid dungeon. In the most extreme example, Sargeras went from a generic bit-part monster to THE Big Bad of the entire universe (albeit a Big Badwho's long dead, as such. Word of God says his spirit is scattered across the Twisting Nether in limbo, but not gone for good).
This has also happened in response to the players falling in love with random NPCs. For example, Hemet Nesingwary was once a minor quest-giver put in as a Shout-Out to Ernest Hemingway. Players either liked or hated him so much that in the first expansion he was moved to Outland and given more quests, and in the second there is an entire faction devoted to fighting him, a third of an area taken over by his crazed former compatriots, as well as yet another group of quests from Nesingwary's hunting party.
High Overlord Saurfang, pre-expansion, served no purpose in the game or the story except to put Nefarian's severed head on a stick and look like a total Bad Ass. He's since become one of the most prominent NPCs in the game. Otherwise, he'll Cleave everyone.
And the Alliance equivalent of Saurfang, Bolvar Fordragon, started out as a stand-in for the disappeared King Varian Wrynn and earned some major cool points for fighting off Onyxia's entire elite guard. When Varian came back, Bolvar played a major role in the Dragonblight campaign, pulled a Big Damn Heroes for you, and died during the Forsaken betrayal at the Wrath Gate.
Nazgrim starts out as a forgettable NPC giving out a handful of quests in one Northrend quest hub, trying to make a name for himself. He seems to have succeeded, since he later appears as the leader of a squad of Kor'kron aiming to lay claim to a new island risen by the Cataclysm. After the mission goes heavily south, apparently his rather clever and valiant efforts to salvage the operation attracted more attention and he's now a general, and leads the Horde (including the player) in Pandaria until the Warchief personally arrives.
Taken to extreme levels with Chen Stormstout and the Pandaren. Chen was originally nothing more than an amusing piece of April Fool's Day art that was beloved by the developers enough to be given a minor role as one of Rexxar's companions in Warcraft III. His general hilarity and his enigmatic homeland were eventually considered interesting enough to be the focus of an entire expansion of World of Warcraft.
Similarly in Star Trek Online, numerous minor characters are revisited, most of whom were mere babies when they were shown on screen. Notable examples include Leonard James Akaar (mentioned above in Television) and Miral Paris, who was born in the last episode of Star Trek: Voyager, and Thomas Riker, Will Riker's transporter double whom the writers of Deep Space Nine cruelly abandoned in spite of fan outcry. There are villainous examples as well; Gul Dukat's half-Bajoran daughter appears, and one of the more enjoyable boss fights is against Gul Madred, the one who tortured Captain Picard.
Because of the nature of Suikoden games all being in the same world, there's usually recurring characters. A number of them are Ascended Extras, frequently as a result of She's All Grown Up. Notable examples are Apple, Futch, Luc, Sasarai, Georg, Lilly, Clive etc. Resulted in a lot of Ensemble Dark Horse.
In Sam And Max Hit The Road, the protagonists may optionally enter the convenience store near their office (which has no bearing on the plot or any puzzles) and save its owner, Bosco, from an armed robber. The interior of the store is never seen and Bosco is never seen nor heard. When the sequel was canceled and Telltale Games was formed as a result, due to his not-legally-a-character status, Bosco was the only character from Hit the Road (besides an appearance of the Human Enigma on a poster, due to an appearance, also on a poster, in the original Sam & Max: Freelance Police comic book series) who could legally feature in their new episodic Sam & Max: Freelance Police series. Indeed, he became a major recurring character, with appearances in all but one episode in the first two seasons and a backstory that is revealed and resolved in Season 2.
While he may seem like one of many other characters the player can pick to go with them in Maniac Mansion, Bernard arguably fits this trope, becoming the hero of Day Of The Tentacle, which suggests that the canonical trio includes him.
In the first Mega Man Star Force, Damion Wolfe and his FM-ian, Wolf, are just Bonus Bosses you can challenge anytime you want. In the third game they were promoted to secondary characters, helping our heroes in an occasion or two. In the anime he was also promoted, but to a "villain" (well, he softens up later).
Mega Man Battle Network has Tory Froid, who is nothing more than a generic, nameless NPC but was upgraded to one of Lan's friends for Mega Man NT Warrior and given a unique design. IceMan is his Navi in the anime but is his father's Navi in the game.
Bass in the first Battle Network was just a Bonus Boss, but became one of the villains in the two next sequels. Battle Network 4 and beyond then turned him into an extra again, but at least he gets dialogue. He was important in the anime and manga adaptations, too.
The weapon salesmen from the first Spectrobes game gets redesigned in the second with his own sprite, cutscene, and name. It's Dave, of all things.
A more notable example would be the Happy Mask Salesman, who, like the Skull Kid, was only part of a minor subquest in Ocarina of Time. He became an unlikely major — and very popular — character in Majora's Mask.
In Ocarina of Time, Anju is just some lady from Kakariko Village who lost her cuccos. In Majora's Mask, she (or an alternate-universe counterpart thereof) and her fiancee Kafei share one of the longest, if not the longest, subquests in the game.
The Capcom vs. Whatever series ascended many minor characters from both Capcom and the rival companies:
Shuma-Gorath, a minor and obscure Doctor Strange villain, is better known as a powerful but unorthodox fighter in the Marvel vs. Capcom games. "CHAOS DIMENSION!" Strange himself did not appear in the series until the Updated Re-release of Marvel vs Capcom 3
Yami is the final boss in Tatsunoko vs. Capcom, and the given explanation why all the characters in that game meet up together. Protagonist Amaterasu had to wait until Marvel vs Capcom 3 for her first appearance.
The Unreal universe has a lot of this. First are Malcolm, Brock and Lauren, who were simple skins a player could choose on the first Unreal Tournament. Then, Unreal Tournament 2003 showed that Malcolm was the winner of the Tournament, with Brock and Lauren as part of his team. Since that moment, Malcolm is considered the face of the Unreal series. Unreal Tournament 2004 and Unreal Championship 2: The Liandri Conflict also ascended Gorge and Arclite, (and later, many others, through a patch) from the same game.
Half-Life 1 has about three "skins" for all of the scientist models, all of whom exist solely to provide exposition, open certain doors and/or die horribly and thereby show the player how not to die horribly. Two of these skins were later used as the basis for two of the main characters of Half-Life 2, which has led to interesting arguments about which of the scientists is the "real" one (Eli is explicitly stated to be the first one to talk to Gordon after the Resonance Cascade; Kliener is generally considered to be the only scientist seen wielding a weapon, appearing at the end of the "Lambda Core" level). Barney, a character similarly based on model skins, this time of the security guards (of which there was only one originally) is more of a Mauve Shirt however.
In Dark Fall: The Journal, Nigel is basically someone you have to rescue. In the spinoff game, The Lost Crown, he is the main character.
Before the final boss of The King of Fighters '94, a blonde woman escorts your team to Rugal's chamber. After a number of victories in KOF '95, a brunette woman informs the unseen Rugal of your team's progress. These two otherwise unremarkable secretaries would go on to become playable characters in KOF '96 under the names Mature and Vice, complete with relevance to the Orochi storyline. And even though they were both killed at the end of the game, they still proved popular enough to return in full-cast compilation games and intercompany crossovers, and may even be rejoining the canon cast in recent games.
Final Fantasy IV: The After Yearsloved this trope—numerous party members were minor characters in the original game, and one such party member was so minor as to be an NPC that only appeared in the ending. Meanwhile, The Dragon ascended from being a one-of enemy in a short sidequest exclusive to the original game's Gameboy Advance release.
Tseng of Final Fantasy VII. In the original game, he is the leader of the Turks, but you never fought him, and he only appears for a short while. Come Crisis Core, he becomes Zack's Turk partner, and could fight alongside him in a DMW sequence.
Vincent Valentine in the original game was just a optional player character. In Advent Children, he's one of the main character along with Cloud and Tifa and he finally got his own spin-off game Dirge Of Cerberus.
While he's not technically a person per se, the Blue Badger in Ace Attorney. He goes from a little plush toy in the first game, to the police mascot and key plot device in the second game, and by Ace Attorney Investigations he's got an entire family and a stage show as well as being the Gatewaterland Mascot.
Also, Gregory Edgeworth. Besides being the deceased father of one of the main characters, and the victim in the DL-6 Incident, he doesn't get much fame...that is, until you play a case as him in Ace Attorney Investigations 2.
Revolver Ocelot, who is introduced in the first game as Liquid's underling—he is the least memorable boss in the game, and has little impact on the plot. By Metal Gear Solid 4, he has become the ultimate Big Bad of the series, and he is the only character to appear in all four games.
In the Total War series, when an army goes into a difficult battle without a general to lead it, and comes out victorious, the army's captain (up to that point a non-entity) can ascend to General status and become a valued member of your royal family (and a powerful unit in his own right).
In the Ys series, Dogi the Wallcrusher was at first a minor character who busts Adol out of jail at one point in the first game and disappears until the end of the second game. However, his bravado and infamous wallcrushing skill instantly turned him into a fan favorite and Dogi became the constant companion of Adol who follows Adol in his later adventures, always busting wall at one point to save Adol in each game. Most recently in Ys Seven, he is a playable character.
Also Anders to a lesser degree, he was a party member in Awakening as well, but his role was greatly expanded on and his character was fleshed out much more in Dragon Age II.
The entire Qunari race. We saw two or three in Origins and heard a little about their culture, in Dragon Age II the plot of a third of the game centers around them. Dragon Age: Inquisition takes it even further, as they join the humans, dwarves and elves as a playable race.
Animated Adaptations of video game franchises seem to be fond of expanding bit players. This was especially common in earlier shows due to the usually basic (or near non-existent) storyline and character involvement in the original material, and thus writers relying on creating extra depths for what cast it has (or in some cases original additional characters):
The SatAmSonic SatAM animated series was intended to develop the initial games "small animal" cast (only generic animals that jumped out of badnik enemies in the game itself) into a team of heroes assisting Sonic. Alterations were made to the core cast in the final revision, however the show still used a redesigned Sally Acorn (aka. Ricky Squirrel in the east) who was arguably the most prominant protagonist other than Sonic himself.
Additionally the Adventures of Sonic the Hedgehog series expanded some of the badnik enemies from the original games and converted them into recurring henchmen for Dr Robotnik. Scratch (based largely on Clucker from Sonic 2), Grounder and Coconuts frequently appeared as comic relief in the show.
This position was reversed around with the promotional SEGA Genesis title Dr Robotnik's Mean Bean Machine, that not only includes said recurring characters as opponents, but uses numerous original badnik characters from the show, the large majority of them background characters from the pilot episode that weren't even given a name or speaking role in the show itself.
And then there's Ada Wong, whose offscreen role in the first game was simply serving as a three-letter name in a memo that provided the solution to a puzzle (a scientist's note mentions that his girlfriend's name is the password). Resident Evil 2 expanded on that bit of trivia to introduce her in Leon's storyline as an important supporting character with a Dark and Troubled Past, and then Resident Evil 4 elevated her all the way up to a full-fledged Dark Action Girl and Leon's Dating Catwoman adversary.
Jiub, a prisoner in The Elder Scrolls III: Morrowind, apparently became a Saint after driving the Cliffracers out from Morrowind. This takes place off-screen, however, and is only mentioned during conversations between NPCs. He makes a cameo appearance in the Dawnguard DLC of The Elder Scrolls V: Skyrim, where he's involved in a sidequest in the Soul Cairns.
Adeleine in the Kirby games may count, depending on if she is supposed to be the same character as Ado or not. Ado was introduced in Kirby's Dreamland 3 as a simple boss character, and Adeleine was one of the main characters in Kirby 64: The Crystal Shards.
Bandana Dee from Kirby Super Star was an extremely minor character, appearing only in the Megaton Punch minigame. In Super Star Ultra, he was given a speaking role and a boss fight in the Revenge of the King arc, and he's playable in Kirbys Return To Dreamland.
Gooey from Kirby's Dream Land 2 was originally a simple health restoring item that sometimes showed up instead of an animal friend, but in Dream Land 3 he was promoted to Kirby's partner in co-op mode (or an optional computer-controlled ally in single-player).
Rick in the Harvest Moon series. In Harvest Moon 64, he was just another shopkeeper. In the PS1Back to Nature titles, however, he was promoted to being a rival slash Love Interest, and has retained this position for the rest of the series.
In another branch of the series, you have Alisa, novice priestess to the Harvest Goddess. IN her first appearance in Island of Happiness, she's was merely the cute assistant to Good Shepherd Nathan. In Sunshine Island, she was one of your potential love interestsnote Though that turned out to be a cruel fake out on the Dev Team's part. In The Tale of Two Towns, she one of two unlockable female love interestsnote with yet again another catch - you can't have a child with her..
Thanks the the experience-based promotion system that Battle for Wesnoth uses and the ability to recall units from previous battles onto the current map, any unit with multiple promotion tiers can go through this in the campaigns, going from "just another generic recruit" to a strong, powerful, and pivotal member of your army.
In fact, this applies to all Daedric lords who've gotten larger roles after Daggerfall. Sheogorath (the only Daedric lord who played a part in the main story of Daggerfall, even if it was limited to showing up in the final dungeon for no readily apparent reason and handing out commentary and riddles) got, as mentioned above, an entire expansion pack centred around him and his Realm, Azura played a key role in Morrowind's main story, Hircine turns out to be behind the Bloodmoon in the Bloodmoon expansion, Oblivion's main story was closely connected to Mehrunes Dagon, and now Molag Bal has been pinned as the Big Bad in The Elder Scrolls Online. Hermaeus Mora also became a very important character in the Dragonborn DLC of Skyrim.
Not to mention Cerberus itself, who go from sidequest baddies to major players in 2 and 3.
Back in the first Samurai Warriors, there's one certain generic officer named Honda Tadakatsu who stood up in defense of Tokugawa Ieyasu during the battle of Mikatagahara, and Takeda Shingen refers him as one of Ieyasu's luxuries. Then Xtreme Legends get announced to have brand new 4 characters, two of them are Toyotomi Hideyoshi and Imagawa Yoshimoto who already got unique character model despite being NPC in the vanilla version, one of them is a brand new archer babe called Inahime, whose father is... Honda Tadakatsu, who is the last playable character and ascended so great he practically became SW's answer to LuBu.
Both The Witcher and its sequel enjoy using this trope. Side characters from the books, such as Triss and Zoltan, get upgraded to main characters in the games, with the former even becoming the protagonist's main Love Interest. Other characters who largely existed as names dropped in the text, like Iorveth, make actual appearances and get fully fleshed out.
Understandable given that the games are officially the continuation of the books now, per the original author.
Butt Monkey Bang Shishigami was treated as a joke by a wide variety of characters, the sole exception being Hakumen. However, an offhand comment by Rachel about an inactive Nox Nyctores turned out to be more than met the ear as Bang ground his ass off and turned into a Minor Major Character as a result - as Chrono Phantasma is moving into Ikaruga, the fans are undoubtedly going to be quite pissed if he doesn't get a significant share of the spotlight this time around. He may still be an over-the-top anime ninja pastiche with a loud voice and a scruffy face, but that's why we love him.
Makoto Nanaya was little more than a bit of Noel, Jin and Carl's memories in CT, but she had significantly changed from the Fun Personified role by the time she arrived in Kagutsuchi in CS. A rumor passed around the internet that she was made DLC in vanilla CS due to plot relevance rather than popularity (not that she isn't popular - far from it), but the full extent came out bit by bit up until Extend - there are reasons why Slight Hope in totality is considered an episode of badassery, and with that degree of involvement in the main story, she is clearly not going away.
Tsubaki Yayoi, Valkenhayn R. Hellsing, and Relius Clover also got the relevance treatment - mere mentions or brief appearances turned into characters with full-blown stories behind them and connections to the plot at large.
Nyarlathotep in the Persona series started out as merely Kandori/Guido's Persona, and wasn't even a final boss. Come Persona 2, it's revealed he is more or less the Personaverse's Bigger Bad, personally responsible for everything bad in 2, both Innocent Sin and Eternal Punishment, and everything that happened in the first game. He's MIA during Persona 3 and Persona 4, but Persona 4: Arena implies that he's the Malevolent Entity, who's still at large post-epilogue.
In the Super Famicom version of Ys IV, Karna is only a minor character that Adol runs into a couple times. However, in the PC Engine CD version, she is a badassAction Girl and a major plot driver.
In the original Little Busters game, Sasami and Kanata exist primarily as Satellite Characters to Rin and Haruka respectively. In Ecstasy and the later sequels, they're fully-fledged love interests with routes of their own.
Name any character from Touhou that was playable at least once. Chances are she debut as a boss character you only see in one or two stage. Notable examples include:
Ice fairy Cirno. She starts out as a Stage 2 boss of Embodiment of Scarlet Devil with no importance to the actual plot at all. Last we heard from her, she gets her own game.
Tewi Inaba. She debuts as a stage 5 mid-boss with no dialogue in Imperishable Night. She's a playable character in the next game Phantasmagoria of Flower View and becomes a main character in the Gag MangaSpin-OffInaba of the Moon and Inaba of the Earth.
Mima also counts, before she disappeared off the face of Gensokyo. In her first appearance in Highly Responsive to Prayers she's just a Stage 10 boss (roughly equivalent to Stage 2 or 3 of later games) and like other characters, have no dialogue. Comes the second game Story of Eastern Wonderland, where she's the Big Bad. She also appears in the third and fifth game as a playable character.
In the firstHalo game, Sergeant Johnson was just a Played for Laughs NPC who happened to be handy with a sniper rifle. He becomes a major supporting character in the rest of the main trilogy, and was one of the protagonists in the novel Halo: Contact Harvest. In part to reflect his growing importance, Bungie went so far as to reveal that he was actually a Spartan-I.
Team Fortress 2 has a habit of ascending characters that appear in promotional material:
Saxton Hale, the Australian C.E.O. of Mann Co., started out as a gag parody of a 1940's Charles Atlas advertisement. The community then embraced him as a Memetic Badass, even giving him a mod where the whole RED team fights him in his honor. Although still not officially in the game, he is a major character in the Mann Vs. Machine arc, hiring the mercenaries after they lose their jobs working for RED and BLU.
In the Australian Christmas update, the justification for the characters being sent in time to DeGroot Keep is that "the Soldier angered a magician." In the next Halloween update, said magician, named Merasmus, is revealed to be the Soldier's roommate (as well as the one responsible for the Demoman having only one eye), and they both hate each other. The Halloween update after that, Merasmus finally loses it and becomes that year's Halloween boss, and this time, he means business.
Sergeant Merrick has a minor NPC speaking role in the original campaign of Dawn of War II. For the Imperial Guard campaign of Dawn of War II: Retribution, he is finally, and thankfully, playable.
Techmarine Martellus goes from the Mission Control in the original game, to a playable character in the Space Marine campaign.
Several of the survivors of the Eldar warhost play roles in Retribution as well. Ronahn and Veldoran are playable in the Eldar campaign, and both were present in the original game, during the Eldar attack on Angel Forgenote Ronahn was the sniper who shouted "Nemerian was the best of us, humans", and takes potshots at your squad from a watchtower, and Veldoran was likely Idranel's second in command Warlock. The female Howling Banshee Exarch who attacks you at Angel Forge is also present as a minor antagonist for one of the optional missions in Retribution.