Affectionate Nickname

Quite simply, when a character gives another character a cute or endearing nickname because the two are either as close as True Companions, or the namer is interested romantically in the namee. The Affectionate Nickname may not be well-received by the one being named, as they find it insulting, but in the mind of the name-giver it's sweet and/or thoughtful.

In short, it's a pet name. Expect The Nicknamer to dole these out a lot.

Examples of Affectionate Nickname include:

Anime and Manga

Naruto refers to his Parental Substitute Jiraiya as "Ero-sennin" (Pervy-Sage). While definitely an Embarrassing Nickname, you can definitely tell it's much, much more about affection as their relationship develops. This becomes especially clear when Naruto still refers to him (when thinking of him affectionately) that way after he is killed.
Yugi is called "Yuge" by his best friend Joey in the Yu-Gi-Oh dub.
In the original version of Yu-Gi-Oh! GX, both Sho and Kenzan call Judai "Aniki", a Japanese term for "big brother". (Meaning that they regard Judai as a Big Brother Mentor.) In the third season, Jim calls Kenzan "Dino-Boy" and calls Asuka "Tomorrow Girl". (It's a rough translation of her actual name; he tends to speak English most of the time.)
Axis Powers Hetalia: Poland calls his best friend/partner Lithuania "Liet". Finland also uses one for Sweden, which was translated as "Su-san" in early scanlations but changed to "Sve" in later scanlations.
Husky and Medley: "Husky" refers to Medley's initial description of her Yuri crush, which included the fact that she owns a Siberian husky. (Thus putting the "pet" in "pet name".)
Fruits Basket gives us Kakeru Manabe of the Student Council who, aside from having a number of nicknames himself, tends to give them to others... whether they want them or not. Case in point: resident Bishounen Yuki Sohma gets saddled with "Yun-Yun," which he points out is a ridiculous nickname since it's longer than his real name. This doesn't faze Kakeru one bit. Later on, he nicknames another member of the student council, "chibisuke." This doesn't go over well either.
In Bleach Yachiru calls Kenpachi Zaraki "Ken-chan" ("Kenny" in the dub). She's probably the only person that can do that.
In Mahou Sensei Negima, Albireo Imma likes to tease Evangeline by calling her "Kitty" (from one of her middle names: Katherine).
In Houshin Engi, Fugen calls Taikoubou, his best friend of many decades, "Bou-chan," or simply "Bo" in the American release by Viz.
Fai just loves making these up for Kurogane. And there's a different one for every occasion!
In Wandering Son, a few volumes in, Kanako gives Saori the nickname "Saorin". Makoto's nickname (Mako-chan) may also count, but he asked Shuuichi to call him that.
Lelouch of Code Geass has always been Lulu to Shirley.
Genkai from Yu Yu Hakusho often refers to her student Yusuke as "Dimwit", whether he actually acts like an idiot or when she's moments away from death. Likewise, Yusuke sometimes calls her "Grandma".
One Piece; Whitebeard was called Oyaji by his crew, a term which, in Japan, means the same thing as "Pops" or "Old Man" when applied to one's father. Luffy also uses this name for Vice Admiral Smoker.
In Lupin III, Oyaji is also what Lupin and his gang call Inspector Zenigata, justified as he's much older than they are, likely at least as old as any of their parents.
The titular Princess Kaguya in The Tale of the Princess Kaguya is nicknamed little bamboo to reflect her abnormal rate of growth.

Comic Books

Batman: Harley Quinn calls The Joker "Mistah J" and "Puddin'." Also, she calls Batman "Bats" or "B-Man" and Poison Ivy "Red". In one animated version (Batman and Harley Quinn) she called Nightwing "Nightwig", her way of teasing him about his hair.)
- In a fanfiction about the Batfamily, Jason calls Damian "baby bird" and "little bird". He has called Dick "Dickie Bird" in canon.
X-Men: Colossus often calls Kitty Pryde "Katya". When he broke off their teen romance after the Secret Wars, he made a point of calling her "Kitty". Later when they resumed their romance during Joss Whedon's run on Astonishing X-Men, he resumed calling her "Katya".
New Gods: The Happily Married Scott Free and Big Barda do this a lot, with Barda's primary nickname for Scott being that he is her "Little Man". Scott actually accepts this nickname whole-heartedly, and even refers to himself as such when talking with her sometimes, especially whenever they're being lovey-dovey.
Superman: Lois Lane has been known to call Clark Kent "Smallville", initially as a form of derision, later as a term of endearment.

Fan Works

The "True Companions" version shows up in Isekai by Moonlight, where the very-tall protagonist Robu-san is called "Oni" by Usagi/Sailor Moon, and he in turn calls her "Bunny-chan".

Literature

Seregil of the Nightrunner series begins calling Alec 'Tali' which means 'beloved.' Guess what happens next...
In V. C. Andrews' Orphans series, the dainty and shy Janet is called "Butterfly" by her orphan friends. Similarly, in the Wildflowers series, the meek and timid Cathy is called "Cat" by her OWP (Orphans With Parents) friends.
- In the Casteel series, Heaven's angelically beautiful and saintly mother Leigh was called "Angel" by Luke after he heard her call her doll that and said that the nickname fit her better. Heaven herself is called "Heavenly" by her brother Tom who's super-close to her (but not romantically interested in her, shockingly enough).
Nicknames are common in L. M. Montgomery's Anne of Green Gables series. Gilbert calls Anne "Anne-girl" after their marriage (after initially calling her Carrots...he didn't make that mistake again!). Anne, in turn, is the only one in the book series to call him "Gil".
- Their youngest daughter Rilla is called "Spider" by her older brother Jem, in reference to her gangly limbs at the start of Rilla of Ingleside. Her brother Walter also takes to calling her Rilla-my-Rilla, a play on her name (she is named after Marilla, though this is actually her middle name). Her childhood friend/crush Kenneth Ford adopts this same nickname, eventually using it to propose ("Is it Rilla-my-Rilla?").
- Jem is always called "Little Jem" by the family's hired help, Susan, which he hates. Susan also calls Jem's youngest brother Shirley her "Little Brown Boy", owing to his dark skin, brown hair, and brown eyes.
In Animorphs, Marco would jokingly refer to Rachel, his (Dark) Action Girl teammate, as "Xena: Warrior Princess." Their Token Non-Human teammate is also known as "Ax" or "Ax-man," which is short for the significantly wordier Aximili-Esgarrouth-Isthill.
- Marco actually does this quite a bit. Jake is usually "Big Guy," while Tobias is usually "Bird-Boy."
In the Elemental Logic series, Zanja calls the family's daughter Leeba "Little Hurricane".
In the Nevermoor series, Morrigan Crow is nicknamed Mog by her patron Jupiter North. She initially despises this nickname, but grows to accept it as the series goes on.

Live-Action TV

Sawyer from Lost nicknamed Kate 'Freckles'.
Dean in Supernatural usually gives nicknames to people he's close to. Sam is Sammy and Castiel is Cas.
On Criminal Minds, Morgan and Garcia call each other a myriad of affectionate nicknames; a common name for him to call her is "Baby Girl".
- JJ is the only person - in the world, apparently - who calls Reid "Spence".
- Morgan, on the other hand, usually calls him "Pretty Boy."
On Bones, Booth is both the one to call Brennan "Bones" and the only one allowed to call her that. She hated it at first, but has come to tolerate it.
- Then there's Daisy, who's nicknamed Sweets "Lancelot".
When Lindsay Monroe/Messer arrived on CSI: NY,Danny quickly dubbed her 'Montana' after her home state. It's not popping up as much now that they're married, though. He's more often hear to just call her 'Linds' or 'Babe' these days.
On Boy Meets World, Cory often calls his Heterosexual Life Partners Shawn "Shawnie".
On Angel, Fred liked to call Wes Bookman during their all-too-brief romance before Illyria came to town.
On House of Anubis, Eddie calls Patricia "Yacker." In a strange twist, this actually started as an insult, then took on a new meaning when they started dating.
In Power Rangers Turbo, Divatox's mother (and leader of their clan of Space Pirates) is never given an actual name, but is called "Mama-D" by her children.

Newspaper Comics

Sally from Peanuts is constantly calling Linus "my sweet baboo." Linus always makes sure to clarify that HE IS NOT HER SWEET BABOO!
- One TV special (with dialogue word-for-word from a strip) has Charlie Brown call the Little Red-Haired Girl this in a letter. "It's a...family expression."

Tabletop Games

The 2nd Edition Dungeons & Dragons accessory The Complete Book of Elves had a version of this. It stated that an elf doesn't keep the name given by their parents when born; they choose another upon becoming legal age. The book also says, however, that even after that, an elf's parents and other elders he or she knew as a child might still call them by the "child name" on occasion, in which case it becomes a sort of Affectionate Nickname.

Video Games

In Grand Theft Auto: San Andreas the Triad clan lead by Wu Xi Mu refer to their leader as their "lucky mole" due to his uncanny ability to participate in gunfights, car races, and video games despite being blind.
In Mass Effect, if you're in a relationship with her, Ashley Williams will refer to Commander Shepard as Skipper.
- In Mass Effect 2, Kasumi will often call you "Shep". Given that she's a fan of you, it definitely fits this trope.
- Male!Shepard calls Tali "Miss Vas Normandy" if romanced. It falls under this because that's not actually her last name (just the name of her ship, which got changed to the Normandy following her trial).

Visual Novels

In Nine Hours Nine Persons Nine Doors, Junpei is called "Jumpy" by his childhood friend June, while June is called "Kanny" by Junpei.
Ace Attorney has nicknames for several player characters, typically made by each character's respective sidekick:
- Phoenix: "Nick", by Maya
- Mia: "Kitten", by Diego
- Apollo: "Polly", by Trucy
Maji De Watashi Ni Koi Shinasai has plenty of nicknames to go around: Gen-san's actual name is Tadakatsu Minamoto, which doesn't come up very often. The whole family calls Shouichi "Cap", short for Captain, and Momoyo, being a year older than everyone else, is "Momo-senpai". Yukie is called Mayucchi by her friends and Mayumayu by Momoyo; Kazuko is known by all as Wanko, but she's also in the habit of doling out nicknames herself - Tacchan for Gen-san, for instance, and Kuri for Chris. This is important later on in the latter two's respective routes, where Yamato starts calling them by their given names instead - in particular, in Wanko's path, Momoyo catches on to their Relationship Upgrade via a sudden lack of this trope.

Web Comics

In Something*Positive, Davan and Aubrey sometimes call each other "Woogie" (Davan) and "Monkeybutter" (Aubrey). A Flash Back series of strips gives the origin of this in their first meeting.

Web Original

Mille of Ilivais X takes to calling Iriana "Ir" early on even before they have any romantic moments, with a justification along the lines of "it's four syllables in six letters!" Iriana has a bit of contempt for it at first but quickly gets over it.
Chaka of Team Kimba in the Whateley Universe often calls Ayla "Ayles". She also calls Hank "Hulkster" (he's a Flying Brick). Ayla calls Jade and Jinn (and Jade's stuff) the "J-Team".

Western Animation

Jimmy of Jimmy Two-Shoes calling Lucius Heinous the 7th 'Lucy', despite Lucius' hatred of the name (and Jimmy).
Inverted in Phineas and Ferb with Jeremy. In the episode The Baljeatles, Jeremy suddenly becomes The Nicknamer and nicknames everybody except Candace. When she finally stops trying to make him nickname her and confronts him about it, he says "I call you Candace, but not because I don't care enough to come up with a cute nickname for you. I call you Candace, cuz, I happen to really like that name." Aww...
Kim Possible is called "KP" by her friend and boyfriend Ron Stoppable.
On Total Drama, Duncan seems to do this for girls he likes—his first girlfriend, Courtney, was "Princess," while his second, Gwen, was "Pasty."
In Doug, the title character's older sister would always refer to him as "Dougie."
On Daria, Jake often refers to Daria as "Kiddo," though he never uses it for his younger daughter, Quinn. This comes among other hints that Jake and Daria are particularly close.
- Similarly, Trent always refers to Jane as "Janie." This could also be used to show their particular closeness (though there's at least one example of their other brother, Wind, using this one too).
Bubble Guppies: Molly refers to Gil as "Gilly". Doesn't help their "subtext" at all.
- On a similar note, Deema calls Nonny "Nonners"...
On Young Justice, Cheshire calls her husband Roy "Red."
On Miraculous Ladybug, Cat Noir calls Ladybug "Milady" and "Bugaboo", although she's often annoyed by the latter one.
Code Lyoko; Odd, Ulrich, and Yumi call Aelita "Princess" and Jeremie "Einstein".
On The Owl House, Luz has called Amity "Cotton Candy Haired Goddess" and "Sweet Potato".

Other Media

The Loch Ness Monster is often called "Nessie" by those interested in cryptids.
- Similar to Nessie, the Lake Champlain Monster is called "Champ" by locals.