>>The Red Priestess had that look like, she’d like to get naked with our boy Jon and apply some leeches. I wonder if he has king’s blood in him like Gendry did. The blood of a Targaryen, perhaps?<<
There is a lot of mystery to who Jon’s mother is. Ned would never speak of it.
And, Martin has kept Jon alive for a long time now - it would seem something big is in store for him.
Do you want rampant speculation? It's not a spoiler because it's just a theory...
Jon Snow is a Stark, but he's not Ned's son. He's the illegitimate son of the "mad king" and Ned's sister, making him half-Targaryan. Ned kept the boy's true lineage a secret - even from Catelyn - to protect his identity.
Dany has three dragons, and accordingly there will be three dragon riders, herself, Jon, and one character introduced in the fifth book and not yet in the TV series. With incest never being a problem for a Targaryan, Dany and Jon will marry to restore the Targaryan throne in the end - the series of books, after all, is a Song of Ice and Fire: Ice (Jon Snow) and Fire (Dany Targaryan) - as well as the white walkers (Ice) vs. the red priests (Fire). A nice duality.
I am currently reading the book series and I have not caught up to the television series. Do you want me to message you and tell you what I know so far about Jon that I have recently learned?