La teoria R+L=J

 the theory makes the argument Lyanna was actually in love with Rhaegar and ran away with him.

After Robert kills Rhaegar thus ending the war, Ned Stark heads to the tower and finds his sister in a “bed of blood.”

With her dying breath, Lyanna forces Ned to make her a promise.

No one knows what this promise was, but many theorize that Lyanna died during child birth (hence the “bed of blood”) and asked Ned to pretend her child Jon was his.

This was done in order to protect Jon from Robert and the Lannisters, who were killing all of the Targaryen children via Ser Gregor “The Mountain” Clegane.

Now, there’s plenty of evidence hidden throughout the story that shows that Jon isn’t Ned’s son out of wedlock.

For example, a quote in the first book shows how Ned is an incredibly honorable man who keeps his vows including those to his wife, Catelyn.

Another piece of evidence is that when Daenerys (who would be Jon’s aunt in this theory) was in the House of the Undying she saw a blue flower growing out of a wall of ice.

Blue flowers are used throughout the series to represent Lyanna while Jon is a member of the Night’s Watch stationed on the giant ice block that is The Wall.

Now, no one really knows if the theory is true or not, but if it is then this would possibly mean that Jon has a better claim to the Iron Throne than anyone else in the story, including Daenerys.