If the prophecy is about riders, Tyrion might be the third
If I’m wrong and the three heads do refer to dragon riders, there are a few options for who might be the third, after Jon and Dany. Bran is one possibility. Recall the OG Three-Eyed Raven’s words to him in Season 4: "You will never walk again, but you will fly." Though this probably just refers to his raven-warging – which we saw in this week’s episode, "Eastwatch" – it could also refer to dragon-warging.
The prophecy could also refer to the potential child of Jon and Daenerys. The Prince That Was Promised theory fits both of them, which could mean their union will secure the next true-born heir and protector of men. Yes, Dany is infertile, but with magic back in Westeros, and the collision of destiny and prophecy, the alchemy may be right for a new Targaryen baby. And remember what Dany called her unborn son with Khal Drogo? The Stallion Who Mounts the World. In Dothraki legend, that’s the name for the savior of men; it’s just another way of saying the Prince That Was Promised or Azor Ahai. Could Dany, like her brother Rhaegar, have had the wrong child in mind?
And then, of course, there’s Tyrion Lannister. In Season 6, he bonded with Dany’s dragons when he released them from their chains in Meereen. There’s a popular and controversial theory in Thrones fandom that suggests Tyrion might also be a Targaryen. His mother, Joanna, was an object of lust for the Mad King, and the timeline works out just right that she may have been in his presence around the time of Tyrion’s conception.
Though the idea of another secret Targaryen is exciting, it would undo Tyrion’s complicated relationship with Tywin, which is a lynchpin of his characterization. Tyrion’s conflicting emotions about siding with an enemy force and his emotional reunion with Jaime become way less interesting if Tyrion is a Targaryen. It also robs Jon’s development of its importance if anyone might be a secret Targ.
But does Tyrion have to be a Targaryen to be a dragon-rider? Sort of. The only known riders in A Song of Ice and Fire were all of the Targaryen bloodline. Dragons were more common in ancient Valryia, where the Targaryens hail from, but all of those families were lost in the Doom, a mysterious event that destroyed the Valyrian Freehold and all who lived there. The Targaryens are the only major family who survived, and thus the only ones still connected to dragons.
Which takes us back to Jon, his past, and his potential future with Daenerys. Together, they’re the basis of all the major prophecies. Only time will tell if it's as individuals or as parents that their fates will be fulfilled.