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 nad 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.