If you believe Jon is Lyanna and Rhaegar's son (which I don't, really, it's a compelling theory, but im not sold on it). Little more than Lyanna forcing Ned to secrecy points to him being their child. If Rhaegar believed he was the prince or was at least one of the three heads of the dragon why would he and lyanna want to keep that a secret? To me, Jon hasn't seemed to be terribly central to the plot IMO (his fate is left ambiguous at the end of ADWD), he could be dead in The Winds of Winter for all we know. To me he was mostly the viewpoint with which to view the problems on the wall, but now there is Melisandre and maybe other character whom we can see the events at the wall through. Even though she is far away from Westeros Daenerys seems to be the one has the most "plot armor," and her story seems to echo Aegon the conqueror.
if he does survive, and is shown to be Rhaegar's son I still doubt he is the prince that was promised. Maybe he is one of the three heads of the dragon along with Aegon VI, if the heads have to be of Targaryen blood.
to me Dany fits the bill of the prophechy more than anyone else, regardless of her bloodline.
Aemon talks about how a dragon's gender is interchangable, and if a Targeryen (who consider themselves actual dragons in human form) is the prince that was promised then it could be either a prince or a princess.
Besides, I doubt his stark honor would allow him to kill Daenerys for a sword, especially if they fell in love somehow. Azhor Azhai didnt just kill anyone for his flaming sword, he killed the person he cherished most in the world (as did Dany kill Drogo when she got her dragons). With how much Dany feels the Starks are to blame for her exile, along with the Baretheons, and Lannisters, I doubt she would warm up to Jon.