Two-Face appeared perhaps three times in the Golden Age, and was unused for roughly twenty years before his Bronze Age revival. This is, however, a somewhat downplayed example; the character's visual was popular enough that there were several stories with "impostor" Two-Faces published over the '40s and '50s, and the character's Golden Age debut was delivered to future generations in several reprint issues.
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 from Batman: The Animated Series who was drawn only because they wanted a girl-coming-out-of-the-cake gag. Joker minions have an average lifespan of less than a single episode, but Harley became big enough to not only jump to the comics, but get her own self-titled series as well.
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.
Much of the human cast of The Sandman. It's 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.
And let's not forget that Jimmy Olsen was a random office boy who eventually graduated to a long-running book of his own as Superman's Pal.
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.
Likewise, James "Rhodey" Rhodes and Carol Danvers were originally just supporting characters in the respective books they appeared in (Rhodey was Iron Man's Black Best Friend and Carol was Captain Marvel's love interest). They eventually became popular enough that they were spun off as superheroes in their own rights, with Rhodey becoming War Machine and Carol becoming Ms. Marvel (and later, the new Captain Marvel).
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!
Similarly with Hawkeye, who started out as a reluctant Iron Man villain, only to become the Avenger with the second most appearances on the team (behind only Captain America), founding leader of their West Coast branch, headliner of the Solo Avengers spotlight anthology, and present in every animated, and now live-action, adaptation of the team.
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.
Squirrel Girl was originally a one-shot character created for a gag story where she teamed up with Iron Man to fight Doctor Doom. She went close to a decade without appearing again until she got revived in 2005, and has since gained enough popularlarity to become an Avenger and get her own ongoing series.
Gwenpool began life as part of a variant cover gimmick that each featured Gwen Stacy dressed as a different Marvel hero (Captain America, Iron Man, Thor, etc.) The character's design proved so popular with cosplayers that Marvel brought her into their official canon as part of the All-New, All-Different Marvel universe — first appearing in a Howard The Duck backup story, she then proceeded to headline a holiday special, before finally getting her own ongoing series.
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.
The Spirit Drinker. In the comics, when D'Ken wanted to dispose of Lilandra, he summoned a soul-destroying critter, it targeted her, Kurt teleported her out of the way, its tongue nailed one of D'Ken's own Mooks, and since it can only take one soul per summon, it vanished. Two panels, notable only 'cause it was the first time Kurt had ever teleported with anyone else (at the time, it was pure agony for him and he passed out.) If someone with a gun had simply tried to shoot her, absolutely nothing would have changed and it comes off as a Big Lipped Alligator Moment. So, in X-Men, the animated series? Well, it was only in one episode, but we're talking one episode of sheer Nightmare Fuel as the Reavers, who'd kidnapped Wolverine to free an alien superweapon, figure out it's a containment unit for something very bad too late, and this only-partially-substantial soul-eating Eldritch Abomination is unleashed to stalk the heroes (and villains through the New York subway tunnels; we get an Enemy Mine out of it) as the faces of half-eaten souls continually form from its substance to cry out.
Psylocke was originally supporting character in Captain Britain comics.
This long and no mention of Earth-616's Blink? Originally, the Earth-616 version was killed off during an alien invasion that leads the creation of Generation X. Blink got a lot of fans when a version of her appear in the famous "Age Of Apocalypse" story. That version of "Blink" went on to become the leader of "Exiles". After twice in failing to restart the "Exiles" business, Marvel Comic had Selene, one of their villains, yank the original Earth-616 version from a pocket universe that the original Earth-616 version was stuck in after killing Harvest.
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.
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 comic's 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".
The Star Wars Expanded Universe is famous for this, taking pretty much any and all minor characters from the films and giving them fleshed out backstories while weaving them into the mythos of The Verse.
Probably the most stunning example is Davin Felth; he's the random stormtrooper who says "Look sir, droids". The comics give him a full origin story, characterization, and show how he realized how monstrous the Empire was before pulling a Heel–Face Turn. He's also shown to have been present at every major event in the first quarter of Episode IV (such as the slaughter of the jawas and Owen and Beru's deaths, which were the Empire's Moral Event Horizon in his eyes) and is partly responsible for Luke and company escaping Mos Eisley, as he fragged his captain during the firefight in the hanger. All of this from an extra with a single line of dialogue.
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.
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.
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.
The 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.
The Transformers: Last Stand of the Wreckers took a bunch of obscure background characters from The Transformers Marvel comics and updated them for their team: Squadron X. Ferak, a redshirt who exploded and died was retconned into being a different redshirt as well and Tornado, another one off character who, again, died in the old run. Now Botcon 2014 is pulling both into their space pirate story, mentioning their Squadron X days, and giving them both toys. Tornado even has a Facebook.
The Transformers Till All Are One: Gives Blast-Off of the Combaticons a central role. While the Combaticons are one of the few Combiner Teams to avert The Dividual Often it was Onslaught, Swindle, or even Brawl who got the character-focus. In the IDW books alone Blast-Off spent most of his time as a background character and was absent from a few of the Combaticon's more important arcs. Here he's the most fleshed out of the Combaticons as it's his machinations that kick off a good deal of the plot and his conscience that prevents more deaths from happening.
Fables had Nurse Spratt, who only had a couple of lines and appearances for the first two-thirds or so of the comic, until she betrayed Fabletown, changed her name and appearance, and revealed herself as one of the most unscrupulous villains in the whole series.
Smithers, the Lodges' Butler, who gets a Day In The Lime Light narrating the fifth issue and backs Archie up on escaping Lodge Manor before the zombies overrun them.
Ginger, a rarely used side character, and Nancy, who has few appearances without her boyfriend Chuck, get quite a few scenes on their own before they join the main group. They're also secretly a couple.
Betty's obscure older sister, Polly, gets a fair amount of screentime in flashbacks.
Despite being made a big deal of in early episodes of Jem, Synergy was mostly treated as a plot-device and rarely even appeared in her holographic form. In the comics there is much more of an effort to present her as a character and as family to The Holograms.
Craig appeared in under five episodes of the cartoon and is mostly treated as "Stormer's brother". In the comics he appears early on and it isn't even revealed he is Stormer's older brother until several issues in. They put more emphasis on his romance with Aja.
Clash is the Misfits groupie in the cartoon but only appears in a select number of episodes. In the comics she appears in almost every issue they do.