Eva, Naomi, Peter, Toby, Tom and Erek in the Animorphs books. Erek is a particularly interesting example: he was introduced in #10 as a one-off character, named after a fan who won a contest. However, he had the perfect skill set to spy on the Yeerks and was also established as knowing that Marco's mother was Visser One's host, making him an easy catalyst for reintroducing her in #15, his second appearance. By #20, Marco (who narrated #10, #15, and #20) considers his dropping in with information more or less routine, and Marco's next book, #25, establishes the plot-enabling concept of having Erek and his friends impersonate the Animorphs so that their families won't miss them if they have to go on longer missions. The very next book, narrated by Jake, has Erek act as a Guest-Star Party Member and the book after that, focuses on rescuing him. He was created as a Character of the Day, but he ends up being so convenient and helpful to both the author and the characters that he was, in the end, probably the most influential of the side characters, facilitating and enabling numerous plots throughout the series and playing a huge role in the final battle.
Neville Longbottom in the Harry Potter books was a fairly minor character not being that much more prominent than Dean Thomas and Seamus Finnegan (who themselves become a little more prominent in Book 5 and 6, and 7 with Dean; though not as much as Neville) until book 5 when he Took a Level in Badass and became much more prominent.
Zigzagged with Ginny Weasley. In then first book she was a Chekhov's Gunman having only one brief appearance then in book 2 becoming essential to the plot. Demoted to Extra in Book 3 and a lesser extent in book 4 then becoming far more prominent in Book 5 onward.
Irene Adler is frequently Promoted to Love Interest or upgraded to a major supporting character in non-canonSherlock Holmes works, even though she only appeared in one original Doyle story ("A Scandal in Bohemia"), and Holmes only briefly encountered her in said story. She makes an impression on Holmes in her brief appearance (understandable, since she's one of the only people ever to outsmart him), but her actual role in the canon series is minimal.
This actually goes for most of the characters people think of as the major cast of the Sherlock Holmes stories. Only Holmes and Watson appear in every story (or nearly every story in Watson's case). Lestrade has only a scant handful of appearances, only slightly more than other police officer characters. Mrs. Hudson is an almost invisible presence (Doyle has her as Mrs. Tuner on one occasion, though this is a writer who had Watson forget his own name). Moriarty appears for the first time in the same story he dies in (then once or twice retrospectively). Mycroft only appears once or twice and doesn't do much. Mary Morstan is pretty much a non-presence after The Sign Of Four. In adaptations these roles are almost always vastly expanded and/or heavily recurring.
"D-list goddess" Khione plays a major role. Gleeson Hedge (referenced once in TLO) is another example. Will Solace and Jake Mason may also apply.
Hylla, Circe's assistant, last seen in Sea of Monsters turns up again as Reyna's sister and queen of the Amazons.
Reyna and Nico di Angelo multiple times - at first Reyna appears in Percy Jackson and the Olympians as one of Circe's assistants, and when Jason remembers details of Camp Jupiter, makes special mention to remember her. Come Son of Neptune, Reyna becomes a major supporting character. Nico, a major recurring character in Olympians returns as a supporting character throughout the series. In the conclusion, Blood of Olympus, both Reyna and Nico are viewpoint characters, the only two narrators who are not one of the seven.
Arwen barely appears in the story proper to The Lord of the Rings, but she has a major role in the Backstory. The films used material from the appendices, which explained her role, to make her a prominent player in the three films.
Gothmog in Return of the King. One mention in the novel (it's never even specified what race he is) becomes a memorable turn in the film with Gothmog as a severely deformed but competent orc general.
The cricket from Pinocchio: in the original story, he was a nameless cricket who was squashed by the title character early on and appeared later as a ghost. In the Disney version, he was given the name Jiminy Cricket and promoted to narrator. Not only did he practically steal the movie away from its title character, but he's gone on to host/narrate other Disney films, short subjects, and even theme park shows. He's also a notable character in several of the Kingdom Hearts games.
Not just the Disney adaptation, though the Disney adaptation surely is the cause. Almost every adaptation will give the cricket (or an equivilent bug) an unreasonably huge role. One way of how telling this is is that one movie took out the Blue Fairy (who didn't give him life in the original but was still a mother figure and was more of a conscience-guide than the cricket) but still kept the cricket character.
Emily Bennett in the American Girl books originally appeared in one book, for two weeks of a story (the Molly franchise) that details almost a year and a half. Since she got her own doll, she's been retconned into more of the story. The Film of the Book does it even more.
A Biography of Tadeo Isidoro Cruz by Jorge Luis Borges is about a secondary character from the epic poem Martin Fierro.
Every character in the film Shrek that wasn't made up for that adaptation is an Ascended Extra, given that it was originally a picture book. This includes Shrek himself, the dragon, Donkey, Lord Farquaad (from the Knight — their specific roles are different, but the parallels are obvious), and Fiona (from the Ugly Princess).
Bosie in Cold Mountain was barely described in the book. He's arguably the most entertaining thing about the movie, becoming a sardonic, acrobatic, sharp-shooting Evil Albino prone to nosebleeds.
Meet the Robinsons is also adapted from a children's picture book, but oddly, only really does this for Mr. and Mrs. Robinson and Louis. In the book, the former two were rather flat characters with little relevance to the strange goings-on the book focused on... and Louis was an extreme example of the First-Person Peripheral Narrator, being a first-person narrator lacking even a name. Most of the other characters are only minorly expanded, or new to this version.
Dune: most adaptations expand the character of Princess Irulan. In the original book, the joke is that she's the author of all the chapter-starting quotes about Muad'Dib, yet only appears right at the end as the Emperor's daughter who Paul marries out of political convenience (and another character remarks that, with such a loveless marriage, she'll have plenty of time for writing...). Which is obviously undermined when the character appears early on, especially in the 2000 miniseries where she meets Paul early in the story and they even seem to have feelings for each other.
Duncan Idaho counts, was well. He has a minor (if significant) part in the first book. He returns in Dune Messiahdespite having died in the first book for a slightly more major part, and by the end of the series, he's practically the only true constant in the Universe. He's been cloned and killed so often that the afterlife probably has a "Duncan Idaho" section. He is the only character to appear in every novel, excepting only the novels that take place before his birth.
In James and the Giant Peach, the rhinoceros that kills James' parents is only mentioned at the very beginning in the book. The Movie, however, gave it a severe upgrade into a major antagonist: it's more of a demonic, lightning-spewing manifestation of James' fears than an actual rhino.
Discworld has numerous Ascended Extras, from Detritus, who started out as a Splatter (like a Bouncer, but he used more force) who didn't even get a line in Guards! Guards!! and eventually worked his way up to being a Sergeant in the City Watch, to Ponder Stibbons, who started out as an inconsequential student at Unseen University and managed to become a senior faculty and later the Archchancellor's right hand man (and by virtue of being the only one of this select group that ever does any actual work, de facto ruler of the University).
Igor is definitely this, starting as a generic gag as part of the whole Hammer Horror/Transylvania/Uberwald schtick and developing by stages into a whole race or clan of meta-humans with all SORTS of medical and scientific abilities, Igorinas, etc
But Discworld's best example is probably Death. He first showed up in The Colour of Magic as a one-off gag, and has since become one of the most major characters in the series, with at least a cameo in every book but one and several stories that star him. Later on, the same happened for the Death of Rats, as it went from a one-off joke to a steady sidekick to Death, with some non-minor roles in most of Death's books.
Terry Pratchett declared that Sam Vimes was planned to be a support character for Carrot, but it just so happened that the whole Watch series ended up revolving around Vimes instead.
Unlike most examples, this came more about because of Carrot's character development rather than Vimes. After his first appearance, one of Carrot's major traits is that it is ambiguous whether he is really a naive, simple person that believes everyone has some good within them or if he is essentially manipulating everyone with this persona. The ambiguity wouldn't work if it was from Carrot's perspective.
The entire Watch as written in Guards! Guards! was intended as a one-off A Day in the Limelight / Take That! to other writers that would relegate them to secondary status. Like Death, though, the Watch proved to be so popular with readers that they became the trope, spawning their own arc of stories. "I wanted to give them a day in the sun, but it turned out to be a full-blown tropical holiday."
The Librarian entered the series as a one-shot gag about one of the many changes the Octavo's spell made in The Light Fantastic. He's gone on to make appearences in almost as many books as Death.
Because of the nature of the Discworld series, characters tends to zigzag in and out of extra/main character. There is no overall narrative, but many different storylines taking place within the same world. So characters like Angua and Carrot will be main characters if the book in question is from the "Watch" subset, but also feature as cameos in non-Watch books. A good example of this sort of thing is Cut-Me-Own-Throat Dibbler, who served as a recurring-gag-cameo in several books before ascending to major character status in Soul Music and Moving Pictures. Then he went back to extra status. Then The Truth had him featuring prominantly again. Whether a character is appearing in a starring role or as a backround person all depends on where Pratchett is swinging the metaphorical spotlight during that book.
Perhaps part of the reason for the trope happening often in Discworld is the high degree of Character Development that occurs, particularly when a contemporary character gets plenty of attention. That's what happened to Vimes—he developed from a dejected Knight in Sour Armor to a justice-minded Genre SavvyCowboy Cop. Death's development was even more dramatic. He started out an emotionless Anthropomorphic Personification but by the time of Reaper Man we see the modern Death of Discworld; he sees his job as Serious Business: the reaper who harvests the old lives to make room for new lives and does what he does so as to maintain order; in so doing, he heels he must also protect his charges from outside influence like the Auditors.
In Lords and Ladies, Agnes/Perdita is just one member of Lucy/Diamanda's teenage coven, albeit the most sensible and a viewpoint character for one scene. In Maskerade she's the viewpoint character for about half the book, and Perdita has gone from being the "witchy" name that all Diamanda's group have to a Split Personality.
Tigger was a minor character in the original Winnie-the-Pooh novels by A.A. Milne, only appearing in The House At Pooh Corner. In the Disney Animated Adaptations, his role was expanded, making him the most prominent character after Pooh himself (and perhaps even topping him!). Roo seems to be undergoing a similar evolution in more modern features (though it is probably worth noting nearly every character has gained a plausible amount of screen time at some point, due to the Disney interpretation's tendency to give A Day in the Limelight).
In The Wonderful Wizard of Oz, Boq is an extremely minor character — a rich Munchkin who lets Dorothy stay the night at his house before she leaves Munchkinland. In Wicked, Boq is a decently significant supporting character during the Shiz University portion, even narrating an entire chapter. Better yet, in the musical, he becomes the Tin Man.
The Good Witch of the North is a relatively minor character in the books, as opposed to Glinda, the Witch of the South. In Volkov's version, where she is called Villina, she is the more prominent one. Added to that, the 1939 movie has the Witch of the North not only take on Glinda's role, but her name.
In Harry Turtledove's How Few Remain, Prussian diplomatic attache Schifflen nearly runs down a careless little girl who charges into the street in front of his horse. In the later Great War and American Empire series, that little girl is one of the primary viewpoint characters.
Several Havenite characters in the Honorverse, with Thomas Theisman the most notable. In The Honor of the Queen, he's a minor, almost forgettable destroyer skipper who appears for a handful of scenes. By Ashes of Victory, he's a linchpin of the series and a major POV character in his own right. Who would have guessed back in his first appearance that he'd change the entire course of the series?
Some theatrical and film adaptations of A Christmas Carol do this with minor or unnamed characters, such as Mrs. Dilber in the 1951 film, and the third Cratchit daughter (given the name Kathy) in the 1970 musical film. Scrooge's ex-fiancee Belle (or whatever she is renamed) usually gets a larger part in the Christmas Past sequence, in at least one version she is even shown in the present, and the 1970 film made her one of Fezziwig's daughters. The 1951 film also includes a heartwrenching scene of Fan's Death by Childbirth. Tiny Tim usually also gets more screen/stage time.
Jeeves himself. In "Extricating Young Gussie", the first story featuring Bertie, Jeeves was merely mentioned in passing a few times as Bertie's valet. According to Word of God, he'd never been intended for a larger role than that, and only became the character we know today in the second story, "Leave It to Jeeves" (later edited and republished as "The Artistic Career of Corky").
Sir Roderick Glossop's wife, Lady Delia Glossop, in the TV series (played by Jane Downs). In the original stories she is an extra, to the point where in Thank You, Jeeves she's said to have died two years ago, with little fuss made over the fact. In the series she never dies, and she's a much more active character who appears almost every time her husband does and has as many lines.
In Larry Niven's Known Space series, the Hindmost is a very interesting variation on this trope. In the original Ringworld novel, he's mentioned in passing, but shows up in the next three sequels (Ringworld Engineers, Ringworld Throne, and Ringworld's Children) as a supporting character. He's a bit of a flat character, however, and we never learn his real name or much of his past history other than he used to rule the Puppeteers. The latter-written prequels Fleet of Worlds, Juggler of Worlds, Destroyer of Worlds, and Fate of Worlds, features as a primary character a Puppeteer scientist named "Baedeker" who later becomes Hindmost of the Puppeteers. The same Hindmost, in fact, who appears in the Ringworld novels.
Zalasta of The Elenium is introduced in a quick scene where he is established as an old friend of Sephrenia's. In The Tamuli he plays a much larger role as an advisor to the protaganists and in the end he is revealed to be the Big Bad behind everything that has happened in Eosia for the last 500 years.
Percy Jackson and the Olympians series has Silena Bauregard who didn't even speak in the first 3 books, and only says 1 line in the 4th suddenly catapults into being a significant and important character in book 5, the last book before the spin offs.
Same could be said of Charles Beckendorf, though his significance only comes around after his death.
Put to extensive use in the Doom series. The cover to the first game featured a second marine running toward Doomguy in the foreground. He was given a Gender Flip and a personality and made into the Deuteragonist, Arlene Sanders. Arlene is Fly's best friend and partner through hundreds of monsters, the moons of Mars, "hell", and the galaxy. But they're only buddies.
The Black Company, first book of the series of the same name, includes The Ten Who Were Taken, ten ancient wizards of great power. By the end of the book, it appears that all ten have been killed, but some survive in secret to reappear later in the plot. Two of these survivors had no speaking lines in the original book, and one of them was never even seen by the narrator, requiring one of the Company Brothers to shout out her name upon discovering her so the reader can realize what's happening.
In Welcome to Night Vale, protagonist duties are shared by Diane Crayton, whose biggest role in the Welcome to Night Vale podcast was a speaking role in "The Debate" episode, and Jackie Fiero, who had one line in one segment of episode 55.
In Spells, Swords, & Stealth, four Non Player Characters from a Dungeons & DragonsExpy tabletop game suddenly find themselves at the center of an incredible adventure when four adventurers get themselves killed in a stupid manner in a local tavern because the players didn't bother reading the rules or listen to the GM. The NPCs, a town guard, the daughter of the town mayor, a gnome with a shady past, and a half-orc bartender, find a scroll that appears to be the summons from the kingdom's mad ruler, who is known to raze entire towns to the ground for any real or perceived slight. They're faced with a choice: if they do nothing, the king is likely to wonder why the adventurers never showed up at his castle and send out a search party which would, eventually, result in the deaths of everyone they know and love. Alternatively, they can hide the bodies and assume the identities of the four adventurers and hope to die far away from their home town to spare their loved ones. In a strange twist of fate, this ends up helping them, when all "real" adventurers find themselves constantly rolling Critical Failures in the dungeon thanks to a Reality Warper artifact that affects not only the "game" world but also Real Life. Additionally, the group also meets three other would-be adventurers who may have found themselves in the same predicament.
The Hunger Games: Gale Hawthorne. He doesn't see much action until Mockingjay with smaller roles in The Hunger Games and Catching Fire.
In the Dreamblood Duology, Mni-inh, Wanahomen, and Hendet — minor side characters in The Killing Moon — become point of view characters in The Shadowed Sun.