Rhaegar wanted his sons to be named Aegon due to prophecy
The Targaryen family was big on prophecy and destiny since they boasted of magical and supernatural elements themselves (namely, their dragons and the unique relationship they had with the magical beasts).
This is elaborated on in the novels that Game of Thrones adapts, where Rhaegar Targaryen was particularly obsessed with the prophecy of The Prince That Was Promised, who may also be Azor Ahai reborn. Aegon is an important name in the Targaryen family because the first Targaryen king was Aegon the Conqueror, who unified six of the seven kingdoms of Westeros (the seventh joining years later on its own terms).
The explanation may be as simple as Rhaegar wanting his son, who he believed to be The Prince That Was Promised and/or Azor Ahai reborn, to be named Aegon Targaryen.
It's not clear if this will be true in the novels, but there is a possibility, considering the showrunners have said that they are following an outline that the author of the novels, George R. R. Martin, has provided them despite taking some liberties. There are some hints to Jon being a Targaryen in the novels too, and perhaps his name even being Aegon.
There is an excerpt in the novels where Daenerys has a vision of Rhaegar discussing with Elia about their son Aegon being, well, the chosen one. This shows how Rhaegar likely believes that the chosen one should be named Aegon, and Rhaegar could believe that his son with Lyanna is this chosen one, not with Elia (or they all are together, but the sons should still have the name Aegon regardless.)
The man had her brother's hair, but he was taller, and his eyes were a dark indigo rather than lilac. "Aegon," he said to a woman nursing a newborn babe in a great wooden bed. "What better name for a king?"
"Will you make a song for him?" the woman asked.
"He has a song," the man replied. "He is the prince that was promised, and his is the song of ice and fire." He looked up when he said it and his eyes met Dany's, and it seemed as if he saw her standing there beyond the door. "There must be one more," he said, though whether he was speaking to her or the woman in the bed she could not say. "The dragon has three heads."