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 we wants to live.
Even Roger seems to be getting slightly tamer. While he still has a few highly malicious moments, he does on occasion show genuine sympathy to the other Smiths (particularly Stan) and even shows capability of learning a Aesop (something he was previously completely incapable of doing). At the very least he doesn't seem quite as outspokenly callous as he was in the third and fourth season.
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).
Dale Gribble from King of the Hill evolved from a crazy, slightly sociopathic Jerk Ass into a still crazy but much more likable character as the series wore on. 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.
Agent Six in Generator Rex, in the episode "String Theory" where the seemingly mindlessly loyal agent actually defused the nuclear bomb rather than allow millions of innocents to die.
Cemented later in "Promises promises" when we see how much he had to do with Rex being accepted by the group.
Even ruthlessly hardened White Knight learned to loosen up in later episodes. So much so that in the finale he freed Six from his duty to kill Rex if the latter's Kaiju form ever manifests.
Due to Heel Face Turns, Kevin Levin in the Ben 10 metaseries, and later Charmcaster. To say nothing of the entire Highbreed species...
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.
Buford from Phineas and Ferb, who was a bully in his first appearance, a Punch Clock Villain in his second, and afterward only makes token attempts to pretend he doesn't like everybody else.
Film historian Leonard Maltin notes that when Mickey Mouse first arrived on the screens, he was hardly a model of good behaviour. Mickey soon evolved into The Everyman, with the Jerk Ass torch being passed to Donald Duck.
June from KaBlam! became this in season four. She still had some cynical moments, but she became a lot nicer.
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.
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.
Rainbow Dash Took a Level in Kindness in "Call of the Cutie" where she eagerly helped Apple Bloom in an attempt to get her cutie mark. Unfortunately, that ended up being the Cutie Mark Crusader's M.O.
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, considering Dash is generally a brash, impulsive speedster while Fluttershy is generally shy and withdrawn from pony society. Compared to Dragonshy, where Fluttershy's weaknesses drove Rainbow Dash crazy, in Hurricane Fluttershy, Dash patiently supported Fluttershy up to and including the very end, regardless of how Fluttershy performed. The Powerof Friendship can greatly influence one's character. 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 and even in The Mysterious Mare-Do-Well when Dash suffered from Acquired Situational Narcissism, Dash developed sort of a Cool Big Sis relationship with her as she was seen affectionately calling her a "squirt" and taking her under her wings. Her heart of gold expanded even 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. Rainbow Dash showed even though she has done "nutty stuff", she has enough standards so as to never endanger anypony's lives. Rainbow Dash showed more of her heart of gold throughout the series.
Her Character Development has been such that 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.
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.
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.
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 basically lower scale versions of 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 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".
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.
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.
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.