Rhaegar Targaryen and Lyanna Stark's son is Jon Snow. That's the popular theory, anyways.
I don't see it. However, I think the formula is still correct. Except it's Robert Baratheon and Lyanna Stark's son.
Think about it, Rhaegar had the typical silver hair and purple eyes. Lyanna presumably had the Stark brown hair and grey eyes. It was well known Robert was in love with Lyanna and he's also a manwhore. They were already betrothed and Rhaegar has two kids with Elia of Dorne. That's not to say Lyanna and Rhaegar maybe didn't have feelings for each other but I think Robert had already knocked her up.
It was no secret that Robert had no love for Rhaegar and visa versa. Stealing off with Robert's soon-to-be-wife sparked the war. She was already pregnant at the time.
We've seen GRRM stress the black over gold in the hereditary department as far as Lannister versus Baratheon goes. Jon Snow is described having grey eyes and dark hair. Could be either Stark hair or Baratheon. The grey eyes goes with the Stark territory.
Now, why hide his identity after the fact? Robert would assume Jon is Rhaegar's bastard. We can assume after the birth Ashara Dayne took Jon Snow down to Starfall until the war ended, where Eddard rode south to give back Dawn and retrieve Lyanna's son to raise as his own. Ashara was close to Rhaegar and potentially loved him. Her "stillborn" child might even have been Rhaegar's child. (As a complete guess, that kid could have been the one the Mountain "smashed" to replace Aegon VI.) Her cover story was a stillborn daughter and threw herself off the tower. I'm wondering if Eddard or Connington didn't deliver the true Aegon to her at the same point and she faked her death and crossed the sea. Still, between her and Connington, I think they both loved Rhaegar more than just friends.