The Ship's Motor

Fanon is an odd thing. Fans come up with it for many reasons. Because Word of Dante says so, because Word of God never said so, because it's the only plausible explanation of an event (sometimes it even is!), or just because it's fun!

One popular form of fanon, however, is that which is created simply to make Shipping the OTP easier. This is the fanon referred to as The Ship's Motor. In the way that having a motor is helpful to a literal ship, this piece of fanon is helpful to a Fanfic writer's ship. But it's not enough for it to be just helpful, to truly be The Ship's Motor, a piece of fanon must have no other clear purpose than to make shipping two characters more plausible. That does not mean the piece of fanon has to be contrived, or even implausible. It can be neutral, or even plausible in Canon. Plausible or not, many of these become well-regarded in the Fandom because so many fans ship the pairing.

Only tangentially related to the Youtube vidcast The Ship's Closet.

Anime and Manga

 * Some Naruto fanfics with the Neji/Hinata couple have the two arranged to marry in order to keep the Byakugan bloodline trait within the family, thus precluding any other ships.
 * Many Neon Genesis Evangelion fanfics with the Shinji/Rei pairing will bring up various arguments and bits of fanon that make the pairing less problematic by reducing the degree of genetic connection between the two. (For the record, as Rei is a genetic construct formed by combining DNA from Shinji's mother and Lilith, in canon she is genetically his half-sister.)

Literature

 * Harry Potter: The Head Boy and Head Girl share a room. Which seems particularly odd since the Founders use magic to keep boys from even visiting the girls' dormitories...
 * There's a million of these, really. There are also "Marriage Law fics," where the Ministry passes a law forcing purebloods and Muggle-borns to marry against their will (usually for the sake of Snape/Hermione or Draco/Hermione). These rarely seem to take into account that she's dating a pureblood in canon and could just marry him. Another plethora of stories has Hermione wind up really being a Pureblood so that she more easily hook up with Malfoy or Zabini. Changing a character's heritage to pair her with an unrepentant Politically-Incorrect Villain? Unfortunate Implications be danged!

Live-Action TV

 * Star Trek: The Original Series: Kirk can empathically sense when Spock is going into Pon Farr.
 * This actually crops up in one of the published TOS books, "Killing Time", which was famously recalled and edited after the publishers realised the author was a Kirk/Spock shipper.
 * Also please note that Gene Roddenberry's novelisation of Star Trek: The Motion Picture stated that it was Kirk's mind that called to Spock across the lightyears and ended his chances at the Kolinahr. Maybe not such a stretch after all...
 * In the Firefly fandom, River/Jayne shippers often use the idea that Jayne's thoughts are "clear" or "simple" as part of the reasons River might be attracted to him. Somewhat more founded than the usual examples of this trope as Jayne's thought process is very simple - the only speculative element is whether River would find that attractive or not.
 * Kara and Lee of Battlestar Galactica are believed to have gone to flight academy together even though canon has discredited this.
 * Faith from Buffy the Vampire Slayer is often portrayed in fanfic as bisexual, building on her on-screen Anything That Moves nature. This is, of course, useful for pairing her with Buffy, Willow, or any other female character. Amusingly, the post-series continuation comics then outed Buffy as the bisexual Slayer. And confirmed Faith as being straight.
 * Fans of The X-Files had a setup for Mulder and Scully's hotel arrangements -- adjoining rooms, door in the middle connecting them -- ostensibly so they could protect each other better from one their Monsters of the Week deciding to attack.
 * Popular: It became fanon for Brooke/Sam shippers that they had an adjoined bathroom connecting their bedrooms, even though there was no such thing.
 * iCarly's Sam/Freddie Double Standard Abuse (Female on Male)/Belligerent Sexual Tension shippers use a reply that Word of God left on his blog as their motor, it's along the lines of "their bickering is a game, and maybe they are afraid to stop playing". The reasonable idea is that any 'game' is their inability admit they really care for each other. The Sam/Freddie shippers take it as the 'game' is wilfully ignoring romantic feelings for each other because they enjoy fighting, and one-day will stop playing and get together.
 * Sam being bi is a big piece of fanon that's very useful in bringing out Carly's If It's You It's Okay feelings for her in the Carly/Sam fandom.
 * Power Rangers SPD: Sky being Bridge's roommate because his hardnosed-ness means his thoughts are more ordered and less apt to bleed over and give Bridge a psychic headache. (Actually, Sky's as emotional as anyone else, if not more. He just feels the need to act like the Perfect Officer™ 24-7 because he wants to follow in the foosteps of his legendary pa.)

Western Animation

 * Like the Buffy example above, Cody on Total Drama Island is often made bisexual, because while it's very clear that Cody likes girls, most fans prefer a bit of Ho Yay between him and Noah.
 * Invader Zim fandom will often justify sex with Irkens by claiming that Irkens have sex at all. Canon seems to imply that they're all grown in a big factory, and naked Irkens have no apparent genitals. It's particularly common to assume that they lay eggs.
 * Also, ZATR fans often portray Zim and Tak having known each other prior to the events on Devastis, even to the point of having been best friends since they were smeets. Based on how they react when they first meet on Earth it seems implied they had never met before.
 * Though Dib/Gaz isn't very popular, those who do ship it sometimes use a Word of God statement about Dib being an experiment created by Membrane to suggest he and Gaz aren't really related. The creator notably never mentioned how Gaz fits into that scenario; since Fanon often makes Dib Membrane's clone, some extend this theory to make Gaz a clone of Membrane's dead wife in a weirdly literal case of Generation Xerox.