I find most theories fun to contemplate. Baby swaps, lineage, true identities. I don't get worked up in any I don't believe because what I do believe can easily be wrong.
It just makes more sense that Jon is the son of a Dayne and Stark rather than a Targaryan and a Stark. Dayne's are still Valaryan. Ned and Brandon both loved Ashara who had dark hair, making it more believable that Jon came out with dark hair. All the Targaryan out of wedlock children (if there were only a shorter word for that the filters didn't *) still had silver hair. Tyrion in the books has silver hair with black streaks in it like most of the out of wedlock children and even the mismatched eyes of Shiera.
But lets think it through. Ned goes and finds Lyanna bleeding out after giving birth. Promises her to keep little baby Targaryan safe. If the baby has Stark hair and eyes, no problem. Ned goes home. But he doesn't go home. He keeps going south all the way to Starfall, to take the Dayne family sword back to his sister, Ashara, who he loves, trusts, and perhaps believes has or had a child of his or Brandon's. I'm not even sure he killed Arthur Dayne. Why would he? For protecting his sister? And the Dayne family LOVES Ned Stark. Why? For killing their son, leaving his bones in a mound of rocks, making their daughter kill herself, but he did bring back that sword, swell guy. But anyway.
So he and Wylla, the wet nurse go off to Starfall way south, like all the way south in Westeros, to return a sword, something Howland Reed could have done. He could have taken Dayne's body too, he was going there anyway, but didn't. Now this child of Ashara's and a Stark, probably Ned's as it would explain a lot more, is pretty likely to have dark hair because Ashara did and Martin is big into hair color showing lineage. So Ned drops of the sword, says sorry for killing your brother for protector my sister, which btw he was in no way obligated to do, and heads home with a baby and no wet nurse. Wylla stays. Did he pick up another wet nurse for the ride home? He is all the way in the south and has to go all the way back to Winterfell. That is a long trip for a baby with no baby dinner. Unless... the baby he took back was older than the one he dropped off. Who is to say when Ned fathered this baby while fighting this war. It could have been 7, 8 or 9 months earlier than Lyanna had her baby. Lyanna's baby and Ned's/Brandon's baby don't have to be anywhere near the same age. Technically Ned could have fathered it before the war, even before the Harrenhall tourney although that is most assuredly where it happened.
So he goes back home to Winterfell and listens to his wife give him *&#( for 13 years about cheating on her. Never mentions to his freaking wife that, hey keep this on the down low but this is really Lyanna's baby, don't spread this around but stop treating Jon like *&#@ because he is the rightful heir to the iron throne. Why wouldn't you tell your wife? Makes no sense to punish Jon, your wife, and yourself for doing the noble thing.
But what if he didn't do the noble thing? He did father a child with Ashara. In one of the episodes Caitlynn does ask Ned about Ashara and he snaps that he doesn't want to talk about Jon.
So Jon is going to be Azor Ahai potentially. Or maybe it will be three people instead of one. But they need a specific sword to fulfill the prophecy. Right now Jon is running around with a nice sword. A Valaryan steel sword Longclaw. But it is the Mormont's family sword. At some point Jorah should end up with that sword. The Stark's family sword has been melted down into Oathkeeper and Widow's Wail. He needs the Lightbringer. If only there was a sword in his family's line that was Valaryan quality but milk white that gave off a dull light, was cool the touch, and made from a comet that fell to the ground 1000s of years ago. One that was named something cool.... like Lightbringer... or maybe something that is the lightbringer... like Dawn. Hey, Ned just returned a sword like that to Ashara 13 years ago. Funny that they need a sword that brings the light, and there is a sword out there named Dawn which is the bringing of the light. If only Jon were heir to that sword.