Fans finally learned about Jon Snow's true parentage in the Season 6 finale of Game of Thrones. In a flashback known to many as "the Tower of Joy scene," viewers saw Lyanna Stark hemorrhaging blood after giving birth. She handed her baby off to Ned, begging him to hide her son with Rhaegar Targaryen from Robert Baratheon, who was head over heels in love with her and likely started Robert's Rebellion over Rhaegar stealing her away. She whispered the baby's name to Ned right before dying, so what is Jon Snow's real name on Game of Thrones? It was impossible to clearly hear what she whispered and it's probable that Ned called him Jon after his buddy Jon Arryn, Lord of the Vale. But why? In part, to hide what would have been Jon's obvious Targaryen name.
As Gilly informed us a few weeks ago, Rhaegar had his marriage with Elia Martell annulled in order to marry Lyanna Stark. One reason for this is probably because he was in love with her. However, another reason for it could be because he believed that one of his children would eventually become the Prince That Was Promised. In the books, the prophecy states that "the dragon has three heads," which means Rhaegar probably thought he need to have a third child for the prophecy to actually come true.
Unfortunately, after giving birth to Rhaenys and Aegon Targaryen, Elia couldn't have any more children. So Rhaegar had their marriage annulled, essentially bastardizing their kids, potentially so that he could have a third son with Lyanna.
Jon Snow was likely given a traditional Targaryen name in line with the Prince That Was Promised prophecy being a Targaryen descendant. One fan theory, proposed by Redditor sparkledavisjr, is that Jon was originally named Aegon, since Rhaegar wanted his heir to have that traditional Targaryen king name. Even though he already had a son named Aegon, that Aegon would have lost his claim to the throne when Rhaegar remarried.
Another, far more popular theory by sparkledavisjr is that Jon Snow's real name is Jaeherys. (The user thinks Lyanna says a three-syllable name in the Tower of Joy scene beginning with "J.") Jaeherys is another traditional Targaryen name of two prior kings. Jaeherys I had a long and prosperous reign over the Seven Kingdoms alongside his sister-wife Queen Alysanne. So the parallels between his marriage and Jon Snow possibly hooking up with Dany, despite their filial relation, are clear. Later, there was Jaeherys II, who would have been Jon Snow's great-grandfather. He had a much shorter (though still good) rule and he was the king who received the original prophecy about The Prince That Was Promised.
It's a general rule of thumb in Westeros to name your heirs after good rulers and avoid the negative connotation of associating them with bad ones. So it makes sense that Lyanna and Rhaegar wanted their baby named after Jaeherys. And we might just have to get over a possible "Daenerys and Jaeherys" coupling.