R + L = J
This is the most notable fan theory of Game of Thrones, which happens before the beginning of the first book, 'Game of Thrones'. Anyway I will run through it from the beginning and explain it clearly (or as clear as I can at least). Back in the reign before Robert Baratheon and the rebellion, the King of the Seven Kingdoms was Aerys Targaryen uncle and father (Targaryens married their siblings commonly) of Daenerys, more commonly known as the Mad King. Anyway, this started simply with roses. Yeah, sounds mad (King, haha I'm joking) but it is as simple as that. In 281 AC (these are of course fake dates, the GOT world is not true), a grand tourney (tournament) was being held in Harrenhal by Lord Walter Whent, in which many Nobles from all over the Seven Kingdoms attended. The most prominent visitor of course being the Mad King, who had not left the Red Keep (his home) for quite some time, but was swayed to by his master of whispers, the enunch Varys. Rhaegar Targaryen, the Mad King's son also attended the Tourney who is the represent R of the theory.
The celebration was a big one indeed, however, the end of it was not as joyous as the rest. At the end of the Tourney, after winning the final tilt (he won the tourney) the Crown Prince Rhaegar Targaryen was to crown the new Queen of Love and Beauty. This is his way of dedicating his win to a woman of his choice, and giving her a crown of roses, this is traditional at the end of all Tourneys. Rhaegar was married by the way, to the Dornish Princess, Elia Martell, but instead of giving her the roses, he rode past her and crowned Lyanna Stark of Winterfell the Queen of Love and Beauty, she obviously the L in the theory. In one of the books of the series, Eddard who was at the tourney recalled this moment as 'the moment when all smiles died,' as Rhaegar was married and giving the rose to a woman is essentially the man's way of saying they wish to court them.
It was an unknown amount of time later when Lyanna Stark was seemingly abducted and raped by Rhaegar Targaryen, who at this point was betrothed to Robert Baratheon. Lyanna's brother, Brandon Stark who was on his way to Riverrun to wed Catelyn Tully, when the news reached him, so chose to go to King's Landing with his compainions. These then lead to Robert's Rebellion in which he over throws the reign of The Mad King, and takes the throne, only to lose Lyanna on the way. However, the theory states that Lyanna wasn't abducted by Rhaegar, but chose to go with him as she was in love with him. It is when she is found in the Tower of Joy by her brother Eddard Stark, dying in a room that smelt of 'blood and roses.' In the books it states she died of a fever, but why would there be blood if she had a fever...
The theory states that Rhaegar did not rape her, but rather they made love together after running away. When Eddard came to find her in the room, she was not weak from fever, rather she was weak from childbirth... In the books, Jon Snow came home with Eddard Stark after he has cheated on his newly wedded bride Catelyn who was promised to his brother Brandon who had died, whilst she was at home with their first son, Robb. One thing that is mentioned from the scene from the Tower of Joy is that Lyanna asked her brother to make a promise to her. Which no-one knows what it was, well, maybe it was to keep her child safe. When Robert Baratheon has taken the throne, he swore he would murder every Targaryen he could, which was when Rhaeger's siblings Viserys and Daenerys were fled across the Narrow Sea where no-one could find them. So of course, Lyanna begged her brother to keep her son safe, and so he did, claiming he was his bastard son, Jon Snow.