When a geeky, stilted or otherwise awkward character steadily becomes more cool, relaxed and well-rounded, this character is brought in to fill the vacuum and emphasize the original's new-found coolness.
Compare Suspiciously Similar Substitute, where the original character leaves before being replaced by a similar one; Over and Under the Top, where two characters start out similar but different in degree; and Cousin Oliver, when a new character is brought in to fill the role of the cute child after the previous one grew up.
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Anime & Manga
In Pokémon, May was brought in as a brand new starting trainer, as main character Ash was becoming a bit more mature and experienced with two regions and the Orange Islands under his belt. This would later be repeated when May and her Suspiciously Similar Substitute Dawn meet in the same Pokémon contest in Sinnoh. However, May hasn't completelychanged...
Archie Comics has Cheryl Blossom, who was introduced as a Ms. Fanservice-type, but also as a contrast to Veronica, who had undergone enough character development that there was some need for a new version of her. Cheryl is bitchier, richer and skankier than Veronica ever was. Veronica, although certainly being rich, bitchy, and seductive, was never outright skanky, but Cheryl clearly was.
In Marijuana Simpson Maggie replaces Lisa as the conscientious daughter, while Lisa focuses almost entirely on smoking pot.
Films — Animation
In Toy Story 2, Buzz was given one of these in the form of a newly unpackaged Buzz Lightyear toy. Like the first Buzz, he had no idea he was a toy. This only served to remind our Buzz Prime how annoying and stuck up he'd acted right out of the box. "Tell me I wasn't this deluded..."
In Warrior Cats, Darkstripe served as the Butt Monkey until he died at the end of the first arc. When his spirit returned to seek vengeance along with the other villains in the fourth arc, Darkstripe had managed to grow stronger because of all the abuse he had suffered and he actually posed a credible threat. Because of this, another villain called Snowtuft was introduced to be the Butt Monkey, and got pushed around and beaten up by the other characters in all but one of his appearances.
Minerva Paradiso in Artemis Fowl book five is a child genius attempting to capture a fairy - in this case not a police officer, but a demon. The similarities are all there between her and Artemis, down to the Greek and Roman God names.
In the second book of The Pendragon Adventure, Spader, a new Traveler, spends the entire book on the same learning curve Bobby had in the first, letting Bobby show off his newfound confidence in comparison. His plot arc even follows some of the same points Bobby's did, down to Uncle Press dying for him the same way that Osa did for Bobby
Angel: Word of God states that Harmony is intended as "Cordelia without the life lessons."
Watcher Rupert Giles (Anthony Stewart Head) became steadily more relaxed, funny, confident, modern and so on, so in time, Wesley Wyndham-Pryce (Alexis Denisof) was brought in to fill the role of wet, panicky, geeky, trapped in the past, obsessed with manners and decorum, uptight, and having no sense of humour. Interestingly, Wesley then underwent a similartransformation as he transitioned from Buffy to Angel, becoming a fully rounded character in time.
Similarly, Anya became this for Cordy, who probably underwent the most amount of character development in the entire Buffyverse. Though in this case, most of Cordelia's character development occurred on Angel, so it was really a matter of her leaving the show and having Anya to fill in that place.
Andrew Wells was this to the Xander Harris of old.
Archeologist Daniel Jackson (Michael Shanks) becomes steadily more physical and cool as Stargate SG-1 goes on (and the writers run out of allergy and geek jokes). During the episode "Meridian", he is rendered unavailable/thought to be dead and is temporarily replaced by an anthropologist, Jonas Quinn who is essentially the big geek Daniel used to be, and remains an ongoing (intermittent) figure of fun.
In one episode Robert Rothman was a geekier archeologist when Daniel was temporarily unavailable.
Dr. McKay starts out as a recurring SG-1 character who shows up whenever there's a need for a scientist to jerkily disagree with the main characters. On Stargate Atlantis, he is one of the main characters, so he gets some character development and Dr. Kavanagh is introduced to take over the "recurring jerk scientist" role.
Sawyer on LOST started out a Jerk Ass and developed into a Jerk with a Heart of Gold. In season 4, he exhibits little (if any) jerky behavior at all, but then Miles was introduced and became the replacement jerk. This is lampshaded even before it becomes blatant:
Miles: Where the hell did they go, tubby? Hurley: Oh, awesome, the ship sent us another Sawyer.
The Mary Tyler Moore Show: When Mary's friend Rhoda got her own spinoff, she brought along her younger sister Brenda to take on the "less cool friend" role.
In Red Dwarf, Rimmer ends up being his own Replacement Flat Character, so to speak: in the TV series, the "original" Rimmer gradually becomes more fleshed out and — if not likeable — understandable. He even shows the occasional moment of bravery, before leaving in series VII to become "the next Ace Rimmer". Then in series VIII he's resurrected by nanobots as his old, pre-hologram self, and is back to being "you as you used to be" as a disgusted Lister puts it.
Meanwhile in the book (and less strongly in the episode "Me^2", the contrast is made between Rimmer and the copy of him that hasn't been "mellowed out" by Lister and co.
After the eponymous star of Chuck grew into a confident, competent and somewhat superpowered spy, his even-geekier best friend Morgan (who had been in the show since the beginning) was brought into the spy team, allowing him to be the same fish-out-of-water adventurer that Chuck used to be.
Frasier was introduced in Cheers as an uptight, prissy intellectual who was usually the butt of jokes. Over time, he developed and even had his days in the limelight; for example, the episode "Love Is a Lonely Snipe-Hunter"). Once he was given his own show, Frasier, and became the central character, he had to become more complex and well-rounded, and his prissyness and bluster were slightly toned down (though still a frequent subject of jokes at his expense). Cue the appearance of his brother Niles, who was everything Frasier used to be and more — geeky, neurotic, and the butt of (almost) every joke.
Part of Sidekick Sergeant Lewis' role in Inspector Morse was being more lighthearted than the title character. When he starred in the spinoff Lewis, he took on a more serious stance and was balanced by his own more optimistic sidekick in Sergeant Hathaway.
Friends had a one-episode example. As time wore on, the originally spoiled selfish Rachel become more complex and likeable (eventually taking over the role of Only Sane Man after all the other characters Flanderized into near self-parody.) To highlight this, a few seasons in, we meet Rachel's sister Jill, who was as spoiled and selfish as Rachel ever was, and highlights how far Rachel has come.
Glee often had to bring in new bully characters once the previous ones become too complex:
For the boys, Puck was the original Jerk Jock. After he joined the Glee Club, Karofsky and Azimio stepped in to fill the bully roles. When Karofsky underwent his own multi-season story arc surrounding his gayngst and Azimio was unceremoniously Demoted to Extra, Rick "The Stick" Nelson was brought in and basically served as a contrast to how far Puck and Karofsky had developed.
For the girls, the Unholy Trinity of Quinn, Santana, and Brittany started off as Alpha Bitches, but all three went through massive amounts of character development and were replaced by Kitty when they (well, Quinn and Santana) graduated. Kitty underwent a Heel-Face Turn by the end of season four, and her former role was filled by Bree in season five.
Jean-Ralphio on Parks and Recreation. Since the first season, Tom's character has deepened and he's been shown as sometimes being good at his government job, and very serious about Rent-a-Swag. So it's been important to hold onto Jean-Ralphio as the lazy, image-obsessed screwup.
Peanuts: Before becoming the blanket-hugging, gospel-quoting weirdo we all know and love, Linus started out as a baby learning to cope with the world. Rerun was introduced to tell those sort of stories again; however Charles Schulz didn't actually embrace the character until well into the last decade of the strip.
On Futurama, Fry's ex-girlfriend Michelle wound up freezing herself after Fry himself was accidentally frozen and woke up in the year 3000, but unlike Fry, who adjusted to the future surprisingly well, she had a lot of trouble with it and eventually hooked up with an also recently-thawed Pauly Shore. Word of God says that one reason they did this episode was because they realized that the original concept of Fry being a Fish Out of Temporal Water not only hadn't really worked out but if anything he functions better in the future than in the 20th century.
In Homestar Runner, Strong Sad has become much more strong and confident, losing a lot of his Wangst and occasionally getting his own back on Strong Bad. Therefore Coach Z and the King of Town have filled in as "pathetic losers" in his place.
Gap: Tom was originally somewhat uptight and nerdy, but eventually became the relaxed character of today, and had his role filled by David.