(Spoilers Extended) Evidence against this theory.
Now, just to be clear, this is not meant to be disrespectful in any way. I just want to voice my opinion and have intelligent debates with other fans. So with that being said, let's continue.
Here are some of the most compelling pieces of evidence that put a certain popular theory to rest permanently and forever,
1.) When GRRM'S wife Parris was approached by a fan at a party who flat-out asked her to her face about whether or not Rhaegar and Lyanna were Jon's true parents, Parris laughed in her face and said,
"Do you really think George would do something as basic as Jon being R&L's son?"
2.) In one of Ned's own chapters, he straight up says that he hadn't thought about Rhaegar in years.
"For the first time in years, he found himself remembering Rhaegar Targaryen."
This point alone could kill this theory forever. Why? Because how can a man knowingly raise another man's son in his house, and not once think about the kid's father at all for years? That makes zero sense.
3.) Jon has no dragon dreams. Plain and simple, he has no dragon symbolism, nor is he connected to the Targaryens in any way. Just wanted to put that out there.
4.) When Ned received word that Robert was traveling up North to Winterfell, he was excited that his old friend was coming to visit. If Jon truly is Lyanna's son, then Ned should've been nervous and panicking, and working to get Jon shipped the hell outta there before Robert showed up, or at least make sure that the kid was kept out of sight. But nope, Ned didn't even have a problem with Jon attending the feast. Granted he was in the back of the hall, but still.
5.) (Edit) Something else I should mention. Ned never once thinks about Jon and Lyanna together in the same category. In fact, whenever Lyanna is brought up, Ned just gets sad and melancholy, but whenever Jon's mother is brought up, Ned gets pissed off and angry. He never gets angry when Lyanna is mentioned.
But what do you all think?