The Ice King was a Parental Substitute to Marceline the Vampire Queen hundreds of years ago, caring for her as a child following her mother's death. While he has since gone completely insane and no longer remembers their history, he still cares about her and she's the only character he has genuine qualms with injuring in any way.
The Lemon Children seem to be this to the Lemongrabs. Normally the earls are rather rude and unpleasant, but they sincerely love their own children — perhaps even more than Princess Bubblegum loves them.
Finn is a downplayed example. While still heroic, Jake still occasionally engages in some really questionable, despicable, or just really boneheaded things when left to his own devices — including stealing, being randomly mean to people, and bragging about his misdemeanors — and only stops being a jerk when Finn reasons with him.
While not really a bad character (all in all), the title character from The Amazing World of Gumball can certainly be a selfish, petty, and greedy jerk who mouths off to just about everyone. But he treats his girlfriend Penny with the utmost caring and respect. He would never hurt her... intentionally. He is also very friendly towards Mr. Robinson despite his blatant hatred and mistreatment of Gumball (even when Gumball and Darwin finally figure out how much he hates them in "The Heart", they forgive him by the end of the episode).
Especially in Season 1 of Avatar: The Last Airbender, the fact that he rates his Uncle Iroh's safety higher than his honor quest is Prince Zuko's primary redeeming quality. There's several easy-to-miss references in their dialogue that show that Zuko, despite his bad attitude, actually does highly respect his uncle's skills and teachings.
Beast Wars, Rampage is a cannibalistic serial killer, akin to Hannibal Lecter as a transforming robot armed with mountain-busting weapons. He openly despises both sides of the conflict and would happily eat them all if he wasn't kept on a short leash by Megatron. However, he finds Transmutate, a horribly disfigured and mentally regressed Cybertronian whom the others view as a useless or even a liability, but as a freak like himself, he sees as a kindred spirit. When Transmutate dies, he actually mourns. As in "collapse onto the ground while weeping loudly" kind of mourn.
Mr. Slinkman, the more level-headed assistant to Scoutmaster Lumpus, is often watching out for him in case of his ill-tempered outbursts.
Due to his crush on her, Jane Doe is probably the only person Lumpus actually shows kindness to.
Mr. Mumbles of Dan Vs. is a more literal example. She's Dan's pet cat and the only thing in his life that he never acts like a Jerkass toward, instead doting on her affectionately. In her debut episode, her Puppy-Dog Eyes persuaded him to adopt her and softened him up enough to release the other animals as he was blowing up their shelter.
Without Jane Lane, the titular character of Daria would be, well, unpleasant to be around.
Also, as seen in the episode where Jane joins the Track team. Daria begins to obviously go mad, voicing her ENTIRE internal monologue.
Darkwing Duck has his adopted daughter Gosalyn, who is probably the person D.W. loves the most other than himself. One episode has her get flung into the future, and her absence from the timeline turns Darkwing Duck into an insane Knight Templar. He still loves her, although now that takes form of locking her up in an fortress so no-one would harm or steal her again.
Webby, a cheery young adventurer, is this to Lena, a sardonic Lady of Black Magic. Webby always brings out the best in Lena, and their friendship is directly responsible for the older girl's Heel–Face Turn.
Goldie is for the most part a selfish, amoral thief, but the only two people she is shown to care about other than herself are Scrooge, (who she has a very long and complicated romance with), and Louie (who briefly served as her protégé).
Sarah actually acts like a nice little girl instead of a miniature cougar when she's around her friend Jimmy.
Edd and Ed are this to Eddy sometimes. While he's not nicer to them than he is to other people, most of his best moments are when he shows just how much he truly cares about the other two.
Mr. Herriman has Madame Foster, his creator, he's normally no nonsense and uptight with pretty much everyone else in the house, including her granddaughter Frankie, but he truly loves Madame Foster (especially when you consider she never abandoned him after she grew up). He's even willing to perform the incredibly embarrassing "Funny Bunny" song for her.
Mac is one for Bloo, as most of Bloo’s nicer moments are either directed at or influenced by his best friend, to the point he was even willing to put his own life in danger in order to save Mac.
In Bender's words, "All those times I said, 'Kill all humans', I'd always whisper "Except one." Fry was that one, and I never told him so!"
Bender occasionally has other, more one-off morality pets. Like the turtles in one episode (Because they both have trouble getting up after being kncked on their back.) and the odd, more serious than a one night stand, love interest.
Zoidberg, despite being the token Butt-Monkey of Planet Express and the trope namer for My Friends... and Zoidberg, is this for Mom and Robot Santa. Zoidberg's the only person to meet Robot Santa's ridiculously high standards and is even offended that Fry and the others didn't think about him for Xmas ("Did either one of you ever stop to think about Dr. Zoidberg?"). Likewise, Mom holds a lot of respect for Zoidberg and even refers to him by his first name, John, and has repeatedly offered him his own staff and medical practice.
On Garfield and Friends in the U.S. Acres segments during the later seasons, Roy Rooster had two Morality Pets. The first one his little niece, Chloe, he ditched his date to save her from the weasel and spent the rest of the night with her instead. The second one is Wade Duck sometimes. Although he pranks him and they argue sometimes, Roy had moments of protective instincts towards Wade. In the "Snow Wade" two-parter, Roy was the only character to show any concern for Wade's well being, when he didn't seem to wake up from the kiss.
Magnificent Bastard David Xanatos has an infant son named Alex in the second season. The Gargoyles helping protect the baby from his god-like grandparents (long story) encourages Xanatos to rethink his priorities.
David's then-fiancee Fox, when she temporarily becomes a werewolf.
Demona's daughter Angela. On any other occasion, Demona is still her usual self.
The kids are this to Grunkle Stan and Ford, especially Mabel. They help bring out the best in them, making up for Stan's greed and Ford's single-mindedness.
Dipper becomes this to Pacifica in "Northwest Mansion Mystery". After saving her from a ghost, she stops being rude and disrespectful to him. His scathing words to her when he found out he was tricked into getting rid of the ghost, and his apology after realizing she regretted her role, allow the two to become friends and a possible interest in each other. When her Abusive Parents tell her to abandon their guests to the ghost, one look at Dipper was enough to get Pacifica to change her ways, making him the first positive influence on her life.
In the He-Man and She-Ra Christmas Special, Card-Carrying Villain Skeletor is chagrined to find himself looking after two Earth kids and their pet who wound up stuck on Eternia.
Hey Arnold!: Arnold Shortman is this to Oskar. Like the rest of the adults at the boarding house, Oskar has a soft spot for Arnold. However, Oskar Kokoshka is also a colossal liar, moocher, lazy bum, and overall Jerkass, and it takes something incredibly drastic for him to want to change. In most of his A Day in the Limelight episodes, it usually happens when he finally drives Arnold so high up the wall, when Arnold is trying to help Oskar be a better person. Oskar usually makes up excuses to Arnold for not working to improve himself or lies to Arnold to get out of work. Arnold loses faith in Oskar and tells him that he's done trying to help him and stomps off, leaving Oskar full of shame. Oskar then makes a genuine effort to make it up to Arnold and show that he can improve himself, restore Arnold's faith in him, and show that Oskar does care about Arnold for all his faults.
Inspector Gadget (2015): In "Back to the MAD Future", Dr. Claw's past self mistakenly believes MADcat was killed at an explosion caused by one of his devices and he decides to become a good guy, creating a timeline where he's Da Chief. Unfortunately, MADcat survives and uses the plans the previous timeline's Claw sent back in time to become such a terrifying villain in the new timeline Gadget and Penny have to go back in time again and restore the status quo.
The eponymous character of Jimmy Two-Shoes is this to Heloise. Since Heloise has a crush on Jimmy, she will only ever willingly do something kind and sweet for him, such as building a funfair or giving Miseryville a snow day. However, not even Jimmy's safe from her Hair-Trigger Temper.
Subverted in the Justice League episode featuring the Justice Lords. Flash is pretty sure that the Mirror Universe Superman (who's gone totalitarian nuts) won't be able to kill him because of what the Flash represented to that group. Reverse!Superman is quite prepared to heat-vision his brains out. His own universe's Superman admits that he knew even Flash wouldn't be able to talk Reverse!Superman out of it.
Flash: Can't do it, can you? I'm the last bit of your conscience. This is the one thing you'll never do.
Superman: I've done a lot of things I thought I'd never do these past two years. One more won't hurt.
Kaeloo: On the rare occasions Mr. Cat does something nice, it's usually for Kaeloo, who he loves and cares about.
Kipo and the Age of Wonderbeasts: Kipo becomes one to Scarlemagne. The two of them are revealed to be akin to adoptive siblings due to Kipo's parents raising Scarlemagne (Or as he used to be known, Hugo), and Scarlemagne begins to genuinely value her friendship and opinion. Unfortunately, it wasn't enough to make him forgo his lust for power, and by the end of season 2 he feels incredibly guilty for betraying her trust, enough to prompt a Heel–Face Turn next season.
Moral Orel: Orel Puppington is this for his grandfather Arthur and possibly for Stephanie.
Fluttershy ends up becoming this for Discord of all beings. She's the first being who he's ever developed a friendship with, and when he realized he'd hurt her to the point she almost broke that friendship, it actually causes him to perform a Heel–Face Turn. In the season 4 premiere when Discord learns the ponies have lost the elements and have no way of stopping him should he decide to go back to his old ways, he still remains good for Fluttershy's sake. In the season 4 finale, when he betrays the Mane 6 and hands them to Tirek, it's Fluttershy's genuine desire to be his friend that makes him reconsider that decision, and after Tirek betrays him he apologizes to her and does a permanent Heel–Face Turn.
Smolder is this to her older brother Garble. Note that while throughout Garble's past appearances in the series he was at best a brute and a bully, and at worst a murderous, sadistic sociopath, Smolder's appearance with him in the episode "Sweet and Smoky" was the first time in the series that Garble is shown being capable of feeling empathy for another living being. Garble genuinely cares for Smolder, and according to her he was a good brother to her in the past. And when Smolder calls Garble out on his actions, he listens to her. She was one of the reasons he reforms at the end of the episode.
One of the first indications that the Golden Guard, Hunter, isn't that bad of a guy is when he meets a small bird Palisman. He at first wants to give it to his uncle, but relents as he bonds with it even working to keep it hidden. It acts as a confident and a beacon of kindness amongst the cold Coven he works for.
Eventually, Hunter himself becomes one for the Abomination Coven head Darius. Even before Darius's more noble traits surfaced, his interactions with Hunter served to humanize him as he tries to get the boy to think for himself and form connections with other kids. Hunter is revealed to be the latest in a long series of clones, with his predecessor being Darius's mentor. Hunter even remarks how sad Darius looked in the past when he took the Guard's mantle.
A literal example; despite Mitchell's coldness, he genuinely loves his dog Cody, and becomes extremely worried whenever he goes missing, such as in "Sunspot's Night Out" and "You Can Call Me Albedo".
Mitchell also clearly loves his mother, and cares about her enough to make a whole detective case out of looking for her missing sunglasses in "Detective Mindy".
Mindylater becomes another one for Mitchell, because she's the very first character to extend a hand in kindness to Mitchell. She's also the only one who knows he's out to expose Jet as an alien, and is able to catch him in the act by sneaking up on him which he finds uncomfortable.
Most of The Park Crew can be pretty big jerks to each other, but they are never (intentionally) mean or harsh to their resident Friend to All Living Things and kooky Cool Old Guy Pops. So much so, that borderline Mean Boss Benson can almost never bring his trademark Hair-Trigger Temper to bare on Pops and Muscle Man (a notorious prankster and probably the closest to being a straight up Jerkass in the cast) swears off pranking when one of his pranks seriously hurts him.
The Park Crew also loves Timmy, a one-shot from "The Heart of a Stuntman". He is a really nice kid whom they watched grow up. Mordecai, Rigby, Muscle Man, and Hi-Five Ghost are actually willing to risk their lives at a Stuntman college to stop his tenth birthday party from being ruined.
In the Samurai Jack episode "Tale of X9", the robot hitman X9, the only one of his model with free will and a processor about him, abandons his job after finding an adorable abandoned puppy, Lulu. As far as he was concerned, all he needed was her... which is why he was forced back to work when Aku kidnapped her.
In the comic book series, Jamshed is this to Apu, especially in the issue "Little Big Mart".
Whenever Bart and/or Homer did something too far, they'll know when their actions upsets Marge. Also, Homer knows he has to fix things if Lisa gets upset, even it means confronting his wife.
Maggie is this to the whole family. No matter how jerkass any of them become in an episode, the one thing they won't do is to hurt Maggie. Even Homer (who often forgets about Maggie's existence) will have his Papa Wolf trait put up to the max whenever Maggie is in danger.
Abe was (by his own admission) a terrible father to Homer when he was growing up and remains abusive toward him now, but he has a special bond with his troublemaking grandson Bart and actively strives to be a positive influence on him.
The titular character from Steven Universe is this to a huge amount of characters: for his Noble Bigot caretakers who have trouble interacting with humans; for Lapis, to who he is the only person besides Peridot that she trusts and cares for; for Peridot, to who he showed the beauty of Earth and taught her to let go of her Lack of Empathy; for Onion, the town's troublemaker who genuinely cares about his friendship with Steven...
While not exactly an amoral person, Kenny McCormick is quite the hedonist, with too much of a taste for sex and drugs for his own good. Once his younger sister Karen enters the picture however, his priorities revolve entirely around protecting her and making her happy.
CraigTucker has developed a soft spot for Tweek Tweak as of the episode "Tweek x Craig". He's gentler and more encouraging towards Tweek than he usually is towards people, and even smiles when around him at the end of the episode. Furthermore, some character notes for South Park: The Fractured but Whole state that Craig only cares about two things: his pet guinea pig (or possibly guinea pigs in general) and Tweek.
Scrappy acted as one to Flim-Flam in The 13 Ghosts of Scooby-Doo. Flim-Flam was a good guy and a teammate to the gang, but also a fairly mischievous child con-artist, Scrappy being the one he was the closest with and the most level with, possibly at least some of it having to do with Scrappy being the one to invite Flim-Flam to join the team at the end of the first episode.
Harry to Norman Osborn in the Ultimate Spider-Man (2012) animated series. Unlike other adaptations, Norman Osborn does seem to care for his son and he shows genuine concern when he hears the Frightful Four are attacking his son's school. He's still strict with Harry, though.
Flug from Villainous is a Card-Carrying Villain and Mad Scientist who loves what he does, that being spreading mayhem and evil through his inventions, but his pride and joy is the genetically-engineered bear 5.0.5., who couldn't hurt a fly; he dotes on him like a devoted father and showers him with boundless kindness and praise, and 5.0.5. loves him just as much.
Kitty Pryde (Shadowcat) becomes this to Lance Alvers (Avalanche) beginning with Season 2 of X-Men: Evolution. While he continues to be antagonistic to the X-Men, particularly Cyclops, Kitty is the one member he has a soft spot for, even going as far as to briefly join the X-Men to be closer to her. She also inspires most of his heroic traits, such as convincing him to to join the battle against Apocalypse during the Grand Finale.
Taken to a dark degree with Young Justice (2010): in the episode "Secrets," Harm killed his younger sister because she was the only person he cared about. This was done in a deliberate attempt to become pure evil.
