Game of Thrones has spent seven seasons so far building a complex world filled with multidimensional characters who have fans coming up with all sorts of theories, ranging from the crackpot to the quite possible. The biggest theory of the show (and books) to date has to be that Jon Snow is the secret child of Rhaegar Targaryen and Lyanna Stark. While that theory hasn't been explicitly confirmed on screen, we can be pretty positive. So, a big question has been what Jon's original name was when he was meant to grow up as a Targaryen. Was it Jaehaerys? Was it Aegon? Was it Rhaegar II? We simply don't know yet. Given the events of Season 7 so far, however, the fact of the matter is that Jon's original name probably doesn't matter much.
We learned that Jon is almost certainly the son of Lyanna and Rhaegar back in the Tower of Joy sequence in Season 6, and Season 7 has progressed to drop more hints that the show is going to reveal his true parentage sooner rather than later. In "Eastwatch," we got two big nods to Jon's Targaryen heritage, and neither was reliant on him having a Targaryen name.
In the first half of the episode, we had Jon getting close to one of Dany's dragons for the first time. Instead of chomping him into King in the North lunchmeat or roasting him like the Lannister army, Drogon stood still and allowed Jon to touch him. It was only the second time that somebody other than Dany got so close to one of the dragons without being killed or scared away, and Dany was clearly shocked that Drogon was behaving for Jon. The scene was pretty clear that Drogon recognized Targaryen blood, even if nobody else aside from Bran has connected the dots just yet. Jon didn't need the Targaryen name to approach Drogon; the blood was enough.
Later in "Eastwatch," Gilly of all people dropped a bombshell about Rhaegar when she stumbled upon a record in the Citadel of Rhaegar's annulment of his marriage to Elia Martell and his subsequent re-marriage, presumably to Lyanna Stark. The record is likely enough to establish Jon as Rhaegar's legitimate son and ahead of Dany in the traditional Targaryen line of succession. There was no mention in the record of a child being born that would give us Jon's original name, but it doesn't matter. There is a document legalizing Rhaegar and Lyanna's marriage, Bran's vision, and Jon's blood that lets him get close to dragons. What's a name compared to evidence that couldn't possibly be faked?
Of course, Jon likely wouldn't be accepted as ahead of Dany in the Targaryen line of succession if he's still going by his Snow surname. If he wants to claim the Iron Throne for himself as a legitimate claimant rather than a conquerer, he'd need to do it as a Targaryen with a full Targaryen name. Then again, Jon has never showed an inclination to take the Iron Throne for himself. Dragon blood or no dragon blood, Jon is a man of the North.
Learning that his father was Rhaegar rather than Ned likely won't suddenly fill him with the desire to sit on the Iron Throne and rule the Seven Kingdoms. He won't need a Targaryen name if he doesn't want to claim the Iron Throne, and the people of the North have already proved that they'll follow his lead, even if he's a Snow rather than a Stark. He was raised by Ned Stark and is a son of Ned Stark, even if his biological dad was Rhaegar. Unless Dany refuses to yield the North to a Snow or Stark, Jon simply doesn't need the Targaryen name.
There is the point that Jon is the most likely candidate for Azor Ahai, a.k.a. the Prince that was Promised. The "song of ice and fire" could pretty perfectly refer to a man of the frigid North with a direwolf who also happens to have the Targaryen blood that makes him a friend to dragons. In the books, Rhaegar intended for his son Aegon to become the Prince that was Promised, and that could be the case in the show as well. If so, given that Rhaegar's son Aegon had already been killed by the Mountain by the time Lyanna entrusted her baby boy to Ned, it's possible that she would have given him the name Aegon in order to fulfill the prophecy espoused by Rhaegar.
If the Prince that was Promised must be a person by the name of Aegon Targaryen, as Rhaegar seemed to believe, then Jon would do well to take the name of Aegon and become the Azor Ahai of the prophecy. That said, Jon might be better off sticking with Jon Snow, because the reality is that Rhaegar wasn't exactly right about everything. Rhaegar's decision to abscond with Lyanna kicked off the events of Robert's Rebellion. The Mad King could have been dealt with in a way that quietly put Rhaegar on the throne without a civil war that ripped apart the Seven Kingdoms; instead, Rhaegar ran off with Lyanna in an affront to Houses Stark and Baratheon, and the Mad King did the rest.
Basically, Jon shouldn't make major decisions based on anything Rhaegar believed just because Rhaegar was his father. He doesn't necessarily need to be Aegon Targaryen due to some prophecy. Game of Thrones has repeatedly proven that prophecies tend to be more guidelines than actual rules of what's to come, and Jon doesn't need to be a Targaryen -- Aegon or otherwise -- to become the Prince that was Promised, if that is indeed his destiny.
At the end of the day, Jon's Targaryen blood and status as Rhaegar's legitimate heir will undoubtedly have a part to play in the Great War and perhaps even what remains of the war for the Iron Throne, but his original name probably doesn't matter. We can continue to theorize, and it should be incredibly satisfying if/when we finally get the reveal of the name he got from his mother and biological father; still, it likely won't have too much impact on the events of the series as a whole.