When Red lady (I still don't know half the names after five seasons) said the blood of kings ran through him, that's when I thought he couldn't be Ned's son.
I don’t think Rhaegar would do that, he loved her. Gave her the blue rose after winning the tournament, bypassing his own wife. Robert claimed that Rhaegar had kidnapped her and kept her in that tower for almost a year, but I don’t think that was the case. I think they were in love and running away.
Ned rescued her from the tower, where she was laying in “a bed of blood”. Her last words to him were ‘Promise me, Ned.’
The only other living person who was aware of that conversation is Howland Reed (Jojen and Meera’s father), who hasn’t been mentioned since Book 1. GRRM says that Howland will make an appearance in the next book....but that could not necessarily be the case.
Also, one of Danerys visions was of a “blue rose coming out of a wall of ice”. The prophecy is that the ‘Dragon has Three Heads’, yet Rhaegar had only two legitimate children with his wife. Jon possibly being the third.
R+L=J is the strongest theory that I’ve seen regarding Jon’s parentage.
I also hold that Jon is Azor Assai reborn (not Stannis, as Melisandre portends).
I’m putting my money on Jon and Danerys ending up together before all is said and done (with Bran able to control the dragons through his warg/greenseeing training)
That was from Robert Baratheon's incomplete (and total anti-Targaryen) perspective. Everyone else who actually knew Rhaegar Targaryen thought he was a great guy, basically the male version of Dany.
The Starks are descended from the Kings of the North, so it's possible that Melisandre is referring to that. Or, she could be giving support to the R + L = J theory.
Personally, I think that Rhaegar and Lyanna ran off together, and the promise that Ned made to his sister as she was dying was to protect her baby, and he decided to do so by lying about the baby's identity.