Gendry Waters. Like most small folk Gendry doesn't actually have a last name, and would only have taken on Waters if Robert Baratheon knew about his existence and formally recognized him as his bastard. In canon Robert never did and Gendry doesn't even know he's Robert's son but most fans refer to him as Gendry Waters anyway, with half the tags on Tumblr and fanfiction putting that as his given name.
You can't throw a stone in this fandom without hitting someone who believes Jon Snow is the son of Rhaegar Targaryen and Lyanna Stark. The fact that the TV adaptation has confirmed it, while the book saga hasn't yet leaves the whole thing in a grey area between Fanon and Canon. Still, how exactly they conceived him is a matter of debate:
The most commonly accepted idea is that Rhaegar fell in love with Lyanna after unmasking her as the Knight of the Laughing Tree and chose her to be the mother of the "third head of the dragon". Lyanna agreed to run away with him because she had feelings for him as well and didn't want to be forced to marry Robert Baratheon, whom she knew would be unfaithful to her.
They could have married in front of a heart tree on the Isle of Faces, and the three Kingsguard at the Tower of Joy might have been there to protect the new heir. But of course there are several variations of this.
Some fans believe the in-universe story, that he kidnapped and raped her to conceive Jon. There's also the possibility that Lyanna consented at first but then changed her mind and Rhaegar refused to let her go.
Related to the one above, many fans take it as a given that the Knight of the Laughing Tree was Lyanna Stark herself. This would also justify how she had Rhaegar got close to each other, since the king sent him to capture the mysterious knight and Rhaegar claimed to have found only "his" shield.
Again, related to the one above is that some believe that Benjen joined the Night's Watch immediately after Robert's Rebellion out of guilt for helping Lyanna run away with Rhaegar, which led to the war.
There are many who believe that Ashara Dayne was actually pregnant with Brandon Stark's child. Word of God stated that Brandon may have left "a little Snow somewhere" and Barristan Selmy's POV in A Dance with Dragons bitterly mentions that Ashara gave birth to a stillborn daughter - and not to Jon Snow, apparently - after Stark dishonored her. Since Barristan has always spoken of Ned with respect and admiration, the only option left is Brandon, who has quite the reputation as The Casanova.
It is also common in fanfics for the eldest sibling to Arthur and Ashara Dayne and parent to Edric Dayne to be a man named Allem.
The idea that Tyrion is Aerys Targaryen's bastard, and thus Daenerys's half-brother. Supporting this theory is the fact that Aerys apparently was "quite taken" with Tywin's wife, that he "took liberties" during the bedding of she and Tywin, and that the passage from the Hand's chamber might have been used by Aerys, not Tywin. Also, Tyrion's hair, in the novels, is a fine blonde, almost white, like a Targaryen. Tywin's line that "I cannot prove you are not mine" is, in the minds of some fans, proof that he suspected or even knew about his wife sleeping with, or being raped by, Aerys.
It's considered practically a given that Young Griff isn't actually Aegon VI, son of Rhaegar and Elia. Fans explain this by saying he's the son of Illyrio Mopatis and his second wife Serra of Lys. What's more, Serra is apparently the sister of Varys, both of them being scions of House Blackfyre through the female line. Only a scant few believe he is who he says he is.
Other fan theories taken as true in some circles of the fandom include Jeyne Westerling, or perhaps her mother bewitched Robb to fall in love with Jeyne. Or the Jeyne Westerling seen by Jaime Lannister in A Feast for Crows and the one seen by Catelyn in A Storm of Swords are two different persons, as their descriptions don't match. This would lead to the speculations that Jeyne is hiding somewhere else and might even be pregnant.
Since "The Rains of Castamere" was only ever given one verse and a chorus by Martin, there's a fan-written second verse that has become popular in fan renditions of the song. At one point, it was so well-known in the fandom that it was erroneously listed with the official lyrics on the series' Wiki.
"Oh, hear my call!" the lookout cried, his eyes upon the Vale, "I see sunlight upon armor, many riders on the trail!" But still, Lord Reyne of Castamere harped on his elegy: "No stripling boy, untried by arms, will play lord over me!" And so he spoke, and so he spoke, the Lord of Castamere, Now rains weep all o'er his walls, with not a soul to hear, The lions at the gate had come, to lift his infant heirs on spears...
Maege Mormont is a lesbian, or her brother Jeor is gay, which is why he went to the wall willingly. Both have children, but that proves nothing, of course.
Or, that Maege slept with Tormund Giantsbane and he sired her daughters. It's unknown who their father is or if she's married. Tormund did claim to have slept with a she-bear and is called "Husband to Bears".
Among fans who don't believe that the series will end with either Jon Snow or Daenerys Targaryen on the Iron Throne, it's taken as a Foregone Conclusion that the series will end with the Tyrells on the throne. Thanks to the series' loose basis in the Real LifeWars of the Roses (with the Starks, Lannisters and Targaryens as stand-ins for the Houses of York, Lancaster and Plantagenet), many history buffs in the fandom have latched onto the theory that the Tyrells are meant to be stand-ins for the ultimately victorious House of Tudor. It helps that the family names are vaguely similar, they're the rather treacherous allies of the Lancaster stand-ins, the Reach's position in Westeros corresponds to the position of Wales (where the Tudor line originated) in Britain, and the family's golden rose sigil looks almost identical to the Tudors' famous red and white rose sigil.
That clever crow of Mormont is commonly assumed by fans to be one of Bloodraven's many aliases, with his unsettling habit of saying the right thing at the right moment especially with Jon Snow around.