Doctor Tomorrowland, currently a member of the Disney-sponsored hero team Imagination in the Global Guardians PBEM Universe, started out as part of the team's background extras. She rose to the rank of Ascended Extra, and then supporting character, and then to featured player, all because one of the players decided to start including her more and more in the story. She later became a full-blown player character.
In Darwin's Soldiers, Dr. Zanasiu's assistant originally didn't even have a name, and was planned to get killed off rather quickly. He didn't, however, and after earning a name (Zachary) he soon became one of the most important characters in the series.
Starting to be a regular thing in the Whateley Universe. One of the campus jokes, the kid with the spirit of the squirrel, got a role in the Phase novels, and then her own story. Side character and campus nuisance Greasy is getting a Christmas special. Big Bad Don Sebastiano got a point-of-view story, and has gotten more screen time. And on and on...
Shinji Ryougi, of Yukari Is Free, originally started as a one-shot gag character to insult a Supporting Character, simply being addressed as Gang Member 122. By the next season, he had elevated to a full-blown member of the Main Cast and actually being referred to by name. The same has happened to other characters, most notably Rei Kakizaki, Kasumi Chiba, and Elesia Rumiko, all of whom started out as extras and were elevated to Supporting Roles and given unique subplots.
Happens all too often in Survival of the Fittest, when characters thought only to be minor characters manage to survive to the end thanks to sheer luck, or simply becoming well-noticed. Two examples:
Lulu Altaire was originally just a member of a four-person posse including Anna Kateridge, Amanda Redder, and (regrettably) Liam Black. Three in-game days later, the death of everyone else she'd met, Lulu becomes one of the final characters alive on the island. She ultimately places second overall in the season, after losing the final battle with J.R. Rizzolo.
AstonBennett was originally only a supporting member of another rag-tag group of misfits. Then, after the group is unceremoniously split up when one of the members turns on the group and is accidentally killed, and the boy she may have had a crush on is murdered right in front of her, she steps up to the plate and sets off on a revenge quest against his killer.
Pretty much the entirety of the cast of Arenas, which in the Quake and Doom universes were just selectable characters for the players.
RWBY: Australian bunny girl Velvet appeared for one short scene in Episode 11, where Cardin is shown painfully tugging on her ears to demonstrate the setting's Fantastic Racism against Faunus and to show what a jerk Cardin is. She very quickly became an unexpected fandom favorite and Monty Oum was so impressed by the fandom's reaction to her that he has promised a larger role for her in volume 2. As of that volume, she's getting her own team, Team CFVY ("coffee").
Fallout Is Dragons is a streamed podcast of a tabletop role-playing group. As would be expected, a significant portion of the cast are NPCs the players took an unexpected interest in and adopted.
Bowser Junior once appeared in an episode of Mario and Luigi's Stupid and Dumb Adventures. He later appeared in the "Bowser Juice" infomercial. He then appeared in "Bowser Junior's Macaroni" and is now a main character, to the point of taking over Mario's position.
Joseph and Cody went from appearing exclusively in the Bowser Junior's Summer School story arc to appearing in most of Junior's late 2014 and 2015 appearances to appearing in every single one of Junior's 2016 appearances (excluding "Bowser Junior's Cookies!" and "Bowser's Goldfish").
The Brooklyn Guy made his debut in a 2011 SML short, and was Put on a Bus the following video. He then returned in part 5 of Bowser Junior's Summer School, and is now a recurring character with over 30 different jobs.
After going through a Chuck Cunningham Syndrome period following his debut, Jeffy reappeared in "Bowser Junior's Game Night 3", and has started to ascend to recurring character position, very much like the Brooklyn Guy did.
Welcome to Night Vale: Intern Chad was an NVCR Intern that everyone assumed died after being sent to investigate a store that was allegedly a front for the world government in a very early episode. In the September Monologues, the Faceless Old Woman's segment revolved around him summoning some kind of Eldritch Abomination that frightens the Faceless Old Woman out of his home (though it wasn't made clear at the time that it was indeed the same Chad). And it later turns out that he is the Big Bad of Year Four, being behind the Strangers and somehow wrapping Maureen into his plans.
Tim from the youtube horror series Marble Hornets is a huge example. He begins the series as merely one of many extra characters who are actors for the "Marble Hornets" independent film being made within the series' own universe. Others include Brian, Sarah, and Seth, and none of these are given much development. Tim only speaks a few random lines in the first few entries, such as Entry #12, and does not stand out, excepting the protagonist Jay's interviewing him of what he knows. However, he becomes even more important in the second series of entries when we find out that He is The Masked Man who has been stalking Jay (presumably under possession by The Operator/The Slenderman). He continues to become a more prominent character in the series, eventually becoming somewhat of a sidekick to protagonist Jay. By the end of the series, following Jay's death at the hands of The Hooded Man, Tim takes over his role as the series' protagonist
In Within the Wires, Season 1's Narrator of the Institute's Relaxation Cassettes, in passing asides about her life, talks about a favorite, underappreciated artist whose work she owns. This painter, Roimata Mangakāhia, is the principal narrator of Season 2, where she analyzes the works of her own favorite artist in museum audio guides.
Entirely Presenting You: Minor side characters like Lawrence and Harrian come back in unexpected, and sometimes devastating, ways.
DSBT InsaniT: Fire Guy is the biggest example out of the entire cast from the earlier eras, because all he did was just exist in an unknown story in the actual DSBT, as revealed in the Special Info Episode.
Robo merely made a last-minute cameo in episode 1, then became a full-fledged character.