The Ice King and The Earl of Lemongrab get these in Adventure Time. The Ice King goes from being a villain to being a mere nuisance who tries to hang out with Finn and Jake because he wants to be their friend. Lemongrab becomes a nicer person after Princess Bubblegum makes him a clone of himself so he won't be lonely and misunderstood- which leads to him releasing his prisoners who, apparently, no longer harbor any ill will towards him. In following episodes, Lemongrab is shown to be a little bit calmer and a little bit less resentful of other people.
Later subverted, however, in the case of Lemongrab. He initially seems to be getting better after his clone is made (though still an asshole), but he takes a hard turn for the Moral Event Horizon in "Too Old".
King Julien XIII in All Hail King Julien. While in other Madagsacar related media, he is self-centered at best (the movies), and an antagonist at worst (Penguins of Madagascar), as the primary protagonist his rougher edges are smoothed out. While he can still be self centered and can suffer from tunnel vision when he sets his mind to something, overall it's shown that he does have his citizen's best interests in mind and does genuinely care about them.
Gumball from The Amazing World of Gumball started off as a Nice Guy but took several levels of jerkass in Season 2. Season 3 actually made him a genuinely nicer kid making him not mind helping others when he usually sighed and said fine in a disappointed manner or snarked against the person in a dickish uncaring matter. There was even a episode called "The Name" that basically extracted and ultimately destroyed his Jerk with a Heart of Jerk attitude! Every episode from there on Gumball basically averts Cats Are Mean on a daily basis.
As of Season 4 he's now a extraneously nice cat child (well the nicest cats in media get) with a massive ego the size of Neptune who you can count on to help you out whether you want it or not and doesn't hesitate to be happy.
The episode ''Frannie 911' revolves around this trope, in which Roger tries to be nicer so he can win back the family's love. However, it turns out that being nice is lethal for Roger's kind, and has to be a jerk if he wants to live.
Rose from American Dragon: Jake Long has this in season 2, following the moment of finding out Jake is the American Dragon by the season 1 finale.
Due to Heel Face Turns, Kevin Levin in the Ben 10 metaseries, and later Charmcaster. To say nothing of the entire Highbreed species...
Valerie Grey from Danny Phantom started out the series as a Rich Bitch who at one point sent Kwan after Danny because he accidentally ruined her sweater. Her dad losing her fortune turned her into a ghost hunter with a nasty vendetta against Danny Phantom. She soon moved past her shallow viewpoint, and became friends with Danny, to the point of breaking off a romantic relationship to protect him. And while her grudge against Phantom never went away, she grew more noble as a ghost hunter even willing to save Dani Phantom, when she found out she was half-ghost.
Razer in Green Lantern: The Animated Series takes several levels in kindness over the course of the series. His love for Aya and the hope in his heart call a Blue Power Ring to him at the end of the series.
June from KaBlam! became this in season four. She still had some cynical moments, but she became a lot nicer.
Kimbo herself, yes the title character of Kim Possible, has this happen to her. During the first season she was fairly bossy and up herself, not uncaring or unfriendly, but something of a hypocrite and would brag about how good she is. Segue to season two and these traits were pretty much gone with her efforts being more It's What I Do and less showing off.
Dale Gribble from King of the Hill evolved from a crazy, slightly sociopathic Jerk Ass into a still crazy but much more likable character by around season four. You can tell by his Vocal Evolution, by the way. Kahn never fully stopped being a condescending jackass, but he did eventually find some common ground with Hank and the rest of his neighbors and came to grudgingly consider them friends of his.
In The Legend of Korra, after losing his bending Tahno initiates a friendly conversation with Korra, despite their previous rivalry.
Lin Beifong was always a Jerk with a Heart of Gold, but during the first part of Season 3 her Jerkass tendencies are taken Up to Eleven due to having to spend time around her estranged half-sister, Suyin. Once they sort out their issues, Lin opens up and becomes a much friendlier person.
President Raiko had always been on the neutral side in Book 2 and had his reasons, but he hadn't been kind to Korra when he banished her from Republic City in Book 3. At the end of Book 3, though, he thanks her for defeating the Red Lotus and welcomes her back into the city. And as of Book 4, he's in better spirits.
In the episode "Hurricane Fluttershy", three episodes after we see Angel Bunny at his nastiest—he's comforting a distraught Fluttershy and helping her improve her flying. While he's back to his jerky self in Just for Sidekicks, this can be attributed to his strong attachment to Fluttershy and the fact that Spike wasn't looking after him.
Rainbow Dash Took a Level in Kindness in the later half of Season 2 as well as Season 3. For example, Rainbow Dash originally had little patience dealing with Fluttershy's meekness, as Dash is generally a brash, impulsive speedster while Fluttershy is generally shy and introverted. In "Dragonshy", Fluttershy's weaknesses drove Rainbow Dash crazy, but in "Hurricane Fluttershy", Dash patiently supported Fluttershy up to the very end, regardless of how Fluttershy performed. Another example is with Scootaloo, whom Rainbow Dash initially dismissed as a "pipsqueak" in "Owl's Well That Ends Well" and treated as a garbage mare. In "Sleepless in Ponyville", Dash developed a Cool Big Sis type of relationship with her as she affectionately called her "squirt", consoled her over her camping fears, and took her under her wing. Her heart of gold expanded more when Rainbow Dash, as daring and bold as she is, admonishes Lightning Dust, a pony with all Dash's jerk but none of her heart of gold, for placing momentary accomplishments over the well-being of her friends and her fellow classmates in "Wonderbolts Academy". In the same episode, she's even willing to give up her lifelong dream to do what's right! By Season 4's "Flight to the Finish" she's been asked by Ms. Harshwhinny herself to coach the young ponies of Ponyville to carry the flag for the Equestrian Games, making her something of a Reasonable Authority Figure. In the same episode, as soon as Dash hears that Sweetie Belle and Apple Bloom left their friend behind, she gently chews them out and rushes back to Ponyville to cheer up a dejected Scootaloo. The Power of Friendship can greatly influence one's character.
Twilight Sparkle starts out as a shut-in who sees other ponies (besides princess Celestia) as constant obstructions keeping her from getting things done. (Studying, finding the Elements of Harmony, that sort of thing.) At the end of the second episode, she's not only realized that friendship is a remarkable magic of its own, but having friends feels good.
YMMV on Trixie. It could be considered that her apology was just to save face. It's not likely she's completely on good terms with Twilight yet. Although her softer personality in the comics confirms her apology might have been sincere. Later in the comics, she asks for Rainbow Dash's help with getting her out of Diamondia and they work fairly well with each other (minus a bit of snarking from time to time) and aids the police and later Applejack, Rarity, Fluttershy, Apple Bloom, and Babs Seed in apprehending a major jewel thief.
Discord after realizing that Fluttershy genuinely wanted to be his friend and he didn't want to risk losing that friendship.
He embraces just how valuable friendship is after Lord Tirek betrays him in the Season 4 finale.
Sunset Shimmer went from terrorizing Fluttershy in the first movie to being consoled by her and cleaning her up in the second.
Starlight Glimmer goes from stealing cutie marks from ponies, imprisoning the Mane Six, and nearly destroying the world due to Time travelling shenanigans, to living in the Friendship Castle as Twilight Sparkle's new student over the course of a few episodes.
Everyone has undergone this development, basically. Benson has been more understanding of the duo after "Benson be Gone" and admits he doesn't hate them in "Busted Cart", and Mordecai only punches Rigby now when he says or does something stupid.
Samurai Jack has Ashi, a young woman who was raised from birth to believe that Jack is evil instead of Aku and must be destroyed. But after she sees the kindness he shows to a ladybug and seeing many people whom Jack had helped, she starts being nicer and becomes Jack's traveling companion and friend.
Though Depending on the Writer, he occasionally snaps back to being nice, then back to a jerk, and so on. This evolution had begun since the show's beginning. Even Season One Homer is a far more sentimental and devoted father in comparison to his counterpart in The Tracey Ullman Show shorts.
And then there's "Alone Again, Natura-Diddily", where Homer promises to treat Ned Flanders less like a Sitcom Arch-Nemesis, complete with a Single Tear (this was in response to Ned's wife dying in a freak accident, so they had to make him act nicer toward Ned).
Sheila Broflovski has undergone this. In South Park: Bigger, Longer and Uncut, she's a Knight Templar Parent who declares war against Canada, and advocates the genocide of anyone with Canadian ancestry, solely because she doesn't want her children exposed to swearing. In later episodes, however, she is much more subdued and open-minded. Compare her actions in the movie to her portrayal in 'Mr. Garrison's Fancy New Vagina', where she scolds her husband for condemning Mr. Garrison's sex change.
Stan and Kyle (and to some extent Kenny) also gained shades of this. In earlier episodes, they were arguably worse than Cartman, obnoxious Bratty Half Pints who had nothing against bullying or manipulating anyone lower on the chain than themselves. As Cartman Took a Level in Jerkass however, they were mellowed out to act as foils to him and the other increasingly insane residents of the town. This is particularly prevalent in their treatment of Butters, while they mistreated him just as frequently as Cartman in season six, as early as season seven they are on more friendly terms and take exception to using him as a patsy for a prank gone wrong.
Eric Cartman himself has come to this territory. If you believe that his actions do see a character so evil and despicable, watch the episode "The Death of Eric Cartman", in it, Cartman seeks self-redemption by believing that he was dead. Another good example of this would be by the end of the episode "Jewpacabra" where he finally accepts the Jews.
Speaking of Cartman, in general, has taken quite a level of kindness from the season 16 and the end of season 15, being less racist and evil, trying to show respect for Kyle in many expectations and even befriending him.
The characters of SpongeBob SquarePants that were Flanderized reverted to their old selves when Steven Hillenburg returned to the helm of the series in general.
Peridot from Steven Universe went from a Fantastic Racist enemy of the Crystal Gems with a severe Lack of Empathy to a more caring, friendly, and open-minded Gem who proudly proclaims herself to be a Crystal Gem and arguing the virtues of Earth to a former ally.
A few other engines have undergone this. Sir Handel and Henry in particular are far less rude and pompous, while James himself has become less snide and bad tempered, evolving more into a lovably vain Mr. Vice Guy.
Scott in Total Drama All Stars is a lot nicer to his teammates and everyone else in general than in Revenge of the Island. It seems getting eaten by Fang made him nicer and dumber at the same time.
While more a difference in interpretation than a canonical evolution in personality, Eeyore in Disney's Winnie the PoohAnimated Adaptations was given a much gentler persona in comparison to his original novel counterpart. This evolved to the point that Eeyore is distinguished as one of the most recognisable Woobies in Western Animation.