First, there's the obvious ones based on jokes/skins. Pantheon is actually a baker when not fighting, there's also some circles which have Pantheon and Morgana running a bakery together, and Pantheon is the more consistently successful one, while Morgana flip-flops between a competent baker or an incompetent one; Fiora is a painter in her spare time (or a fan or artwork in general). Ahri is a part-time Idol Singer with a penchant for backstage antics. Akali is actually a trained nurse (to a lesser extent, also Shen and Kennen being a surgeon and doctor, a small circle believes Surgeon Shen is the one responsible of visual updates). Alistair having a milking fetish. Taric being taken from his Ambiguously Gay in canon and just being full-blown camp, and often being The Bear to Ezreal's twink. Darius being the Only Sane Man in the upper echelons of the Noxian command hierarchy, though this might be because of his brother and the other.... colorful personalities there. Evelynn being a bigger vamp than even Ahri. Tryndamere having a bad habit with his right arm. Xerath is a fan of/interested in LARP'ing. Valor being a really naughty bird or a mini-Troll to Quinn.
Then there are the ones that spawned from forum jokes. Talon is an alcoholic and a Dogged Nice Guy towards Cassiopeianote The Alcoholic part is because of Cassiopeia being a bit of a whiny brat. Vladamir's Pungeon Master status being taken Up to Eleven, where there are some comics/fanfics where he can't go a single sentence without a pun being somewhere in there, there's also a Running Gag with him being an excellent sower because of his pants looking like they were made from a Circus tent. Gangplank's Bad Boss tendencies being also taken Up to Eleven, sometimes he goes out on a raid and needs to hire an entirely new crew because all of them died; either from enemies or by him killing them. Wukong mustn'trunaway. Xin Zhao will make a manout of you. Mordekaiser cannot laugh normally, but instead his laugh is always going to be "Huehuehuehuehuehuehue". A fan idea for a skin created the meme of Yasuo being a Sashimi chef.
There's a very common theory that Xin Zhao isn't a Demacian native, but is actually from Ionia. This is mainly because he's one of, if not the, only Asian-inspired champions not from Ionia. Furthering this is that he has a mutual friendship with Sona, who spent her childhood in Ionia and is staying in Demacia at the moment.
Fans of the lore generally believe that the first name of General du Couteau (Katarina and Cassiopeia's Disappeared Dad) is Markus.
One fan-held belief that is equal parts amusing and kind of disturbing is that, of the three yordle girls, only Poppy is Hidden Buxom under her armour. Kind of makes sense though when you consider that, as the only yordle of EITHER gender to be a melee-based bruiser (not counting Rumble, who rides a Mini-Mecha, or Gnar who's only a melee bruiser in Mega-Gnar form) she'd be a prime candidate for Boobs of Steel.
The two common Braum fan theories: He's an Iceborn. Or he is a manifestation of the people's ideas and general hope in the Freljord. Thus on some level making Braum immortal so long as people continued to listen and believe in the stories of him. The latter theory essentially makes him Runterra's Badass Santa. Which is awesome.
The fan theory that Garen's sword is magic, and is the source of all of his non-physical abilities. The blade itself is the catalyst for Judgement, and the hilt is what creates the shield for Courage.
Speaking of Garen and later Lux, it's also fanon that Garen is excessivelyoverprotective of his little sister, to the point of scaring Ezreal away from pursuing a serious relationship with her. Not much has been officially revealed on how Garen feels towards Lux, except as a stoically understanding big brother that believes she is a strong girl on her own and has enough willpower to make her own decision. He may even welcome Ezreal's companionship with Lux. As for Garen himself, he is mainly depicted as a professional Demacian soldier whose attraction to Katarina is an interesting conflict of interest.
Each skin having its own backstory (if drastically different from the base skin's appearance/lore) is pretty common as well. For example, Underworld Twisted Fate is from a timeline where Twisted Fate got his powers by making a Deal with the Devil; Justicar Syndra is the version of Syndra who understood her elders' concerns and used her talents for good; Atlantean Syndra being a member of Fizz's race; The Commando line as essentially The Squad with some of the Demacian Champions, etc.
There's a really weird bit of fanon involving Lissandra. Apparently she hides fluffy cat ears underneath her helmet. This is actually fairly common in fan art and no one really knows where it started from. But it has spread very quickly. Example: here◊.
For some reason, almost every serious picture of Thresh as a human before he died gives him a Glasgow Grin. While being unknown where it started, many fans who know of it agree: If a Thresh skin of him as a Human is made it needs the sliced mouth. Except the gender-flipped ones.
Before she was even released, people already started making Kalista into an Overly Attached ADC due to her kit's theme. Yandere Specter anyone?