Agent Phil Coulson is one of the most triumphant examples ever. He started out as a minor comic relief character in the first Iron Man movie. His popularity led to his role becoming recurring in Iron Man 2 and Thor. Then he became the main character of two of the Marvel One Shots short films. Then he got a role in the Ultimate Spider-Man animated series and became a Canon Immigrant to the main 616 universe in the comics. Then he got a much bigger role in The Avengers, even appearing on some versions of the poster. And now he's Back from the Dead and is starring in his own series. All within five years since his first appearance.
"Happy" Hogan, Tony Stark's bodyguard in Iron Man played by director Jon Favreau, has considerably more lines in the second movie; in addition, he gets to beat down a guard, ram Ivan with his car, and has Black Widow (aka Scarlett Johansson) put him in a headlock with her legs. He has a smaller but important appearance in Iron Man 3 (where once Happy is hospitalized, Tony declares war on the villains), only to resurge in the Spider-Man movies.
Iron Man 2: Jack White was originally hired as the food stylist who prepared and served the salmon carpaccio to Vanko in the aircraft hangar, but he ended up appearing in several scenes as Jack, Hammer's assistant.
In the comics, Peggy Carter is, at best, an important footnote to Steve and his primary love interest, Sharon Carter. In the films, though, she's become one of the most prominent supporting characters, even after Captain America: The First Avenger seemingly concluded any more adventures in the 40s. The first issue of the 2011 Captain America series even uses her death from old age as a jumping-off point specifically because she'd now be familiar to audiences. It's worth noting that the filmmakers have tried to use Peggy in every single (Earth-bound) MCU vehicle after The First Avenger. Joss Whedon wrote a scene for her in The Avengers, and she has cameos in Captain America: The Winter Soldier, Agents of S.H.I.E.L.D., Age of Ultron, and Ant-Man, plus, she took center stage in Agent Carter, which was such a huge hit that it spun off into her own regular TV series. Even the first episode of What If...? has an alternate version of Peggy known as Captain Carter as the main protagonist.
In the comics, the Iron Legion were an extremely minor group that only appeared in two issues back in the 90's. In the MCU, they have pivotal roles in both Iron Man 3 and Avengers: Age of Ultron.
Hope Van Dyne is an extremely minor character from the comics, and only appeared in the Marvel Comics 2 continuity. Here she's promoted to major character status and is effectively the film's female lead. She becomes the new Wasp in The Stinger, and even gets her name in the title of the sequel.
Likewise, Darren Cross doesn't have the same scale of villainy in the comics as he does as the movie's Big Bad. He's also MUCH cooler looking in the movie.
Hawkeye, the Avenger that got the least amount of screentime in the previous movie, gets a considerably larger role in the sequel. His family is even briefly shown.
J.A.R.V.I.S. in a sense, he ascends from an A.I. created for the MCU to become The Vision.
Dr. Helen Cho. In the comics, she's an extremely minor Posthumous Character, and is only notable for being the mother of Amadeus Cho. Here, she's more fleshed out and takes Phineas Horton's place as the creator of the Vision.
Laura Barton. She was a very minor character in The Ultimates who only appeared in two issues, and is most notable for dying to advanceHawkeye's storyline. She is not only alive in the movie, but has a fairly sizable role and gets some great lines.
Kraglin had barely a handful of lines in Guardians of the Galaxy and accomplished no more than the nameless Ravagers. In Guardians of the Galaxy Vol. 2, he plays an important part in both Taserface's mutiny and Yondu's subsequent escape, and effectively becomes the Guardians's getaway pilot during the fight with Ego. He also has a large role in Yondu's funeral.
Nebula's role as The Dragon in the first Guardians of The Galaxy film was relatively minor; she had maybe a dozen lines, and was there primarily so Gamora had someone interesting to fight. She has an expanded role in Guardians of The Galaxy Vol.2 — and then becomes one of the main ensemble in Avengers Endgame, and the catalyst for the third act.
Maria Rambeau is an extremely minor character in the comics, but plays a major role in the Captain Marvel movie, where she's Carol's Heterosexual Life Partner.
The Stark Industries scientist William Ginter Riva (played by Peter Billingsley) only appears in one scene in Iron Man, in which Obadiah Stane yells at him for not being able to miniaturize Tony Stark's arc reactor. In Spider-Man: Far From Home, he's now Mysterio's chief programmer.
In Guardians Of The Galaxy, Cosmo The Space Dog is seen in a non-speaking appearance as part of the Collector's collection, escaping the captivity in The Stinger of said movie. The appearance seems not to be more than an easter egg for comic book fans as this dog neither speaks nor shows his comic book powers. Cosmo returns in The Guardians of the Galaxy Holiday Special - now revealed to be female in the MCU - in a speaking role actually interacting with the Guardians and revealing her telekinetic powers. She's also set to return in the upcoming Guardians Of The Galaxy Vol. 3.
Live-Action TV
Most of the characters in Agents of S.H.I.E.L.D., by necessity. Because almost any character with any real prominence in the comics has been earmarked for use in the films or Netflix shows, AOS largely has to deal with obscure barrel-scrapers who never got much page time in the comics. For instance, despite only ever appearing in just over a dozen or so comic issues before the show, Lance Hunter is one of the main characters.
While Mariah Dillard is among the main villains of season 1 and in 10 of 13 episodes, she's always forced to play second-fiddle to the main villains of Cottonmouth and Diamondback. In season 2, she is the main villain overall front and center, and the season is about her trying to stay on top as others like Luke, Misty and Bushmaster try to bring her down for various reasons. The same is true of Shades, who also gets more screentime in season 2.
Shades' cellmate / prison lover Darius "Comanche" Jones only appears in a flashback episode of season 1 about Luke's time as a prisoner at Seagate. In season 2, he has a much more prominent role.
Sugar, an enforcer for the Stokes' and later Mariah's driver, has a more prominent presence in season 2 compared to season 1.
D.W. Griffifth only gets a few scenes in the first season, first being seen peddling HD footage of the Incident to Harlem passerby outside Pop's, and later being used by Luke to get information on Domingo Colon's base of operations. In season 2, he's a more prominent fixture, having become Luke's videographer and now selling Luke Cage-branded merchandise out of Pop's Barbershop.
Arthur Harrow was an extremely minor Moon Knight villain in the comics, having only appeared in a single issue. He was promoted to the Big Bad of Moon Knight, albeit as a Composite Character with Sun King and very little in common with his comic counterpart.
was originally hired as the food stylist who prepared and served the salmon carpaccio to Vanko in the aircraft hangar, but he ended up appearing in several scenes as Jack, Hammer's assistant.