One of the longest-running mysteries of Game of Thrones was finally resolved in Season 6 with the reveal that Ned Stark is not Jon Snow's father. The Tower of Joy flashback revealed that Lyanna Stark gave birth to the baby boy, then entrusted him to her brother before she died. While the show has yet to confirm that Rhaegar Targaryen is Jon's father, the odds are pretty great that Jon Snow is the product of Rhaegar and Lyanna, which could make him the rightful heir to the Seven Kingdoms rather than Daenerys. There are people who likely know that Jon is (presumably) Rhaegar's, including Howland Reed, who has yet to appear in the show.
Assuming Rhaegar + Lyanna = Jon, the big question is whether or not Rhaegar and Lyanna were married after they ran off together. If they were, Jon is a legitimate Targaryen and ahead of his aunt Dany in the line of succession. If not... well, then Jon is still a bastard. Jon's position in the Seven Kingdoms at the moment is relatively precarious and would be much stronger if he was acknowledged and legitimized as a Targaryen. Dany isn't likely to legitimize a man raised in the North whose claim to the throne is better than her own out of the kindness of her heart, and the word of a heretofore unseen character like Howland Reed might not carry much weight. There is one event that could definitively prove that Jon is a legitimate Targaryen heir, and that event is the on-screen wedding of Rhaegar and Lyanna.
Admittedly, Rhaegar was already married to another woman. Princess Elia of Dorne was in King's Landing with their two children when the city was sacked. That said, the Targaryens were historically proponents of multiple spouses. Rhaegar could have easily taken Lyanna as his wife without violating his marriage to Elia, and his children with both women would be legitimate. Rhaegar was also far saner than his father, so he probably would have known to marry Lyanna after he ran away with her. Absconding with the only daughter of a high lord was bad enough; getting her pregnant out of wedlock would be worse. After all, he likely wasn't planning on a civil war to ensue that would result in the end of Targaryen rule in Westeros. Rhaegar and Lyanna were probably married.
A few seasons ago, Rhaegar and Lyanna's marriage might not have made much of a difference. Flashbacks were rarely used in Game of Thrones, and Cersei in Season 1 proved that words on a piece of paper aren't good for much when the Iron Throne is at stake. Thanks to Bran's abilities as the Three-Eyed Raven, however, it's possible for a present-day character to witness the wedding of Rhaegar and Lyanna that resulted in Jon. Bran already knows that Jon is Lyanna's son; it would make sense if he delved further back into the history of that era to see how Lyanna ended up in her position.
Of course, Dany and many others would need to actually believe Bran if he claimed that Jon is the secret son of Rhaegar and Lyanna. For people who haven't been following these stories on page or on screen, the whole idea would probably sound preposterous. Still, if Bran can prove his abilities, Dany and Co. might not be able to deny his claim that Jon is a Targaryen. Maybe Melisandre or Thoros of Myr could see it confirmed in their flames.
It's not out of the realm of possibility that Game of Thrones managed to film a wedding scene without spoilers leaking. Lyanna Stark was already cast back in Season 6, and there's a chance that Rhaegar was cast as well. A Reddit user connected some dots that could point toward Welsh actor Wilf Scolding as Rhaegar in Season 7. Back in September, Aisling Franciosi, who plays Lyanna, began following Scolding on Twitter, and he followed her back. At the time, he also started following the official Game of Thrones Twitter account and actor Liam Cunningham.
Obviously we can't definitively say that Wilf Scolding is playing Rhaegar Targaryen based on his Twitter activities back in September, but it's something to go on. Game of Thrones takes concealing spoilers very seriously, and the casting of Rhaegar would be a huge spoiler. The Twitter follows might be significant clues. Then, there's also the fact that Scolding is blond with light eyes. He's not quite Daenerys/Viserys/Aerys blond, but he could pass for a Targaryen.
If Wilf Scolding is playing Rhaegar in Season 7, a wedding scene between Rhaegar and Lyanna would make sense. There aren't that many episodes left in the series, and showing the two characters getting married would settle the question of Jon's parentage. It wouldn't even matter that he and Dany look nothing alike. She's full-blooded Targaryen with generations of incest to her name, and Jon is half Stark. Who knows? Maybe Game of Thrones will reveal that Tyrion is also a secret Targaryen, and Dany will realize that there must be a third to ride the third dragon. As long as Jon is willing to renounce his claim to the Iron Throne, Dany might be glad to have another family member and inclined to let him keep the North. That, or marry him in grand Targaryen tradition.
We'll have to wait and see. There's no way that Jon's heritage as both Targaryen and Stark won't be a factor in the upcoming Great War. Now that Bran is back in Winterfell and Jon will have to head back to the North to prepare to fight the White Walkers, the truth may come out sooner rather than later, especially if Dany comes up as well with her firepower, assuming Cersei doesn't get somebody to shoot them all out of the sky.