What the the prophecy is likely referring to
Prince Rhaegar, son of the Mad King, brother Daenerys, and instigator of Robert’s Rebellion, is a fascinating character we know only through the memories of those who knew him. Though he lives in infamy as the "kidnapper" of Lyanna Stark, his friends – and even some of his enemies – remember a softer soul who played his harp, sang to the lowborn, and was fiercely intelligent, noble, and quietly tender.
Rhaegar was also obsessed with prophecies, to his own undoing. He believed that his direct bloodline would produce the Prince That Was Promised, a hero who would save the world of men from the next Long Night. Rhaegar was married to Elia Martell, sister of Oberyn, and with her produced two children: a daughter named Rhaenys and a son named Aegon. At first, Rhaegar believed Aegon was the prince that was promised, but had an apparent change of heart after learning his sickly wife could have no more children. Another ancient Targaryen prophecy spoke of the return of dragons, specifying the dragons would have "three heads." Though it’s never spelled out in the text, one can assume that -- given Rhaegar’s words in Dany’s vision -- that he believed the two prophecies were related.