Much of the human cast of The Sandman. Its traditional to introduce a character in one story arc as a minor background character, only to have them reappear in a later arc as the protagonist.
Unity Kincaid. In "Preludes and Nocturnes", she briefly appears as one of the victims of the sleeping sickness. In "The Doll's House", she is revealed to be the grandmother of that story's protagonist, Rose Walker.
Barbie. In "The Doll's House", she's one of the guests at Hal's boarding house. She later turns out to be the protagonist of "A Game of You", which follows her adventures in the Dreaming after she breaks up with her boyfriend Ken.
Martin Tenbones. First appears in one frame in "The Doll's House", where he's one of the creatures in Barbie's dream. He appears as a living being in "A Game of You", when Barbie travels through her own dreams.
Lyta Hall. Briefly appears in "The Doll's House" as a prisoner of Brute and Glob. Then Dream vows to take away her child, and...things get more complicated. After a few sporadic appearances in later issues, she becomes the protagonist of "The Kindly Ones".
Daniel Hall. Introduced as Lyta Hall's baby, who Dream vows to take away when he's old enough. After a few background appearances, he plays a central role in "The Kindly Ones". And in "The Wake", he becomes the new Dream after the original's death.
Before becoming the superhero Black Goliath (later just Goliath) and getting his own series, Bill Foster started off as Hank Pym's lab assistant in a few issues of The Avengers.
John Constantine, star of Hellblazer, was originally just a recurring extra in Swamp Thing who looked like Sting. Artists Steve Bissette and John Totleben convinced writer Alan Moore to give him a speaking part, just because they liked drawing Sting so much. His popularity took off from there.
X-Men has quite a few characters whose role is expanded in adaptations:
Bolivar Trask, creator of the Sentinels. In the comicverse, he dies in the three-parter that introduces them, in a Heroic Sacrifice once they get out of control. His son and his nephew also tried their hand at it, but also didn't stick around. Despite X-Men's love of Back from the Dead, Bolivar took forty-three years to get there. He finally came back from the dead in the main Marvel Universe in X-Force v3 #6. In the intervening years however? In the 1990s animated series, he's a Recurrer you can expect to see in many a Sentinel story, on the run from his own creations. In X-Men: Evolution, he's arrested after the initial (unauthorized in this version) Sentinel incident, but kept around by SHIELD to design more once Apocalypse comes onto the scene. Had the series continued past the Apocalypse arc, we'd no doubt have seen a lot more of him and his wayward Humongous Mecha children. He's got an expanded, recurring role in Ultimate Marvel as well.
Berzerker of the Morlocks (yeah, with a Z). Comics: seen in one issue. Sympathetic but completely nuts. His friend Scaleface is killed by the cops because of Cyclops destroying their cover (he figured it'd make 'em stand down and hadn't counted on the cops shooting first and asking questions later) so he goes, well, berserk and dies when he's knocked into water during the ensuing battle (frying him with his own electrical powers). In X-Men: Evolution, though, he's an ex-Morlock and one of the newer students, with the show for three seasons. Temperamental but not Ax-Crazy.
Scaleface herself. In the 1990s series and Evo, her power to turn into a big nasty lizard really gets to shine. Though she doesn't see an increase in screentime, her role goes from "dies so Berzerker can go even more nuts" to "Morlocks' resident bruiser." Berzerker wasn't even in the 1990s series.
Karl Ruprect Kroenen from Hellboy. In the comics, he appeared briefly in Seed of Destruction, then figured in a minor subplot of Wake the Devil, at the end of which he died. In The Movie, he had as much screen time as the main villain, Rasputin, and his death was pretty ambiguous.
Oroku Saki, aka The Shredder, was originally a one-time villain. However, he grew popular with the comic's readers. The rest is history.
In the original Mirage comics Chet was the name given to the turtles' original owner, whose panel time can be counted on one hand. In the IDW comics, however, he is now one of Stockgen's head scientists, and April's supervisor.
Sunstorm of Transformers. In The Transformers, before the Decepticon roster grew enough to fill large battles with known characters, repaints of Starscream were often used when Megatron needed more Mooks. One of these repaints, seen very briefly in the premiere, was bright yellow. The comic books flesh out this blink-and-you-miss it extra into a radioactive berserker, and he's still getting toys and other appearances, and is homaged in Transformers Animated (the kiss-up Starscream clone has his coloration.)
Another recolor, this one lime green and had no speaking part in the episode that originally featured him, was given the name Acid Storm and had a few appearances in comics and his own figure. He even got an homage of his own in the Aligned-continuity character of the same name.
Many, many Transformers have a role that's much larger than that of previous holders of the name. For example, Red Alert was a background character in a few episodes of G1, and his one A Day in the Limelight episode had him go insane due to damage. Transformers Armada and Transformers Cybertron make him The Medic and a central character. Also, Overhaul was previously a Redshirt in the Dreamwave Comics series: his first panel involved him getting a hole blasted through him.
Transformers: More than Meets the Eye: Gives big parts to characters unimportant in all other adaptations: Like Swerve and Tailgate who debut as a group of Autobots shooting at Trypticon, here they're the Comedy relief team of a Motor Mouth Bar tender and a bot who's spent almost all of his life trapped underground having missed the war. Characters like Chromedome and Brainstorm, while important in other continuities, did almost nothing in the IDW series until this comic came along. Whirl was just a Wrecker with an interesting Cycloptoid design, he rarely had much characterization and died in many of his old appearances. Here he's become the Heroic Comedic Sociopath who's hated by all his co-workers and may have accidentally started the war by beating up Megatron.
This happens to at least two characters in the film adaptation of 300; Stelios turns from a teenage Spartan who is ridiculed by the others when he tires out to an adult, battle-hardened soldier who exemplifies the Spartan image, and the Captain's son is given a name (Astinos) and takes Stelios's place as the eager young Spartan.
Two-Face appeared perhaps three times in the Golden Age, and was unused for roughly twenty years before his Bronze Age revival;
The Riddler and The Penguin made more frequent appearances, but were still, at best, recurrent characters. Today, they are regular cast members. The Penguin is now an unshakable Gotham crime boss; Two-Face, the Riddler, and Man-Bat have all Ascended to "dubious ally" status.
Catwoman is now firmly an Anti-Hero, and rather less "anti" than many of her peers in that group.
Harley Quinn was a one-shot Joker minion drawn only because they wanted a girl-coming-out-of-the-cake gag. Joker Minions have an average lifespan of less than a single book, but Harley became big enough to have a self-titled series.
Sandra and Monique from the Scott Pilgrim series. Lampshaded in the final volume when they get multiple introduction scenes.
Parodied in the very first issue of Bartman #1 during the scene where Bart Simpson, Milhouse Van Houten, and Martin Prince all meet in the comic-book shop. Bart points out one comics panel (unseen to the reader) featuring a street scene with a random pedestrian in the background. Bart explains that the writers plan to kill this guy off in the next issue and then bring him back to life as a supervillain known as "The Jaywalker"!
In Spider-Man, Eugene "Flash" Thompson was once a Jerk Jock who bullied Peter Parker, but was a major fan of his web-slinging alter-ego. After the shift away from high school, Flash tended to stick to the background before becoming a soldier. After an accident cost him his legs, he got a second chance in serving his country. He is now the newest Venom, a member of The Avengers and dating Valkyrie. Now, that's impressive.
The Black Widow was just a supporting character for many years, not a full Avenger (with good reason — she started out as a Russian spy, and this was the Cold War). She's now been an Avenger long enough that it's hard to think of her otherwise, and adaptations reflect this. In Avengers Assemble she's one of the main cast, and the movie The Avengers bumps her all the way up to founding member!