This is probably the most popular one out there right now. Within the books, there does appear to be plenty of evidence supporting this. It is mentioned numerous times that Eddard made a promise to Lyanna on her death bed, but in all 5 books it has never been revealed. Martin makes a point of telling the story of Lyannas capture, rape and concealment in the Tower of Joy by Prince Rhaegar. In various snippets by multiple characters he goes on to tell the tale of Eddard and Howland Reed (father to Meera and Jojen) and several others as they attempt to rescue Lyanna. They ultimately fail, as Lyanna dies there in pools of blood, and only Eddard and Howland escape alive. They also make a point in the books of mentioning how much Arya looks like a young Lyanna, and how Arya and Jon Snow bear a strong resemblance, also lending credence to this possible plot line. If Jon Snow turns out to be the son of Rhaegar, it also follows the story, as he would be half Targaryen and make up the third head of the dragon, as was foretold to Daenerys. It makes sense that Ned would keep Jons true parentage a secret to protect him, since Robert swore death to all Targaryens, even going so far as to claim the child as his bastard, and suffer the shame. So there you have it. 10 strange and wonderful theories that may or may not be true. Which one do you think is most likely?