Way back when, the Targaryen family mastered the rare art of dragon-riding through a mix of sorcery and good old-fashioned practice. In Daenerys's case, she seems to be able to ride them without the use of magic; she's shown to have bonded with the creatures on an intense level (she's their mother, after all).
So, as previously stated, Daenerys seems like a given in regards to the prophecy. Not only is she already riding Drogon on the regular, but she's the one who hears the prophecy in the House of the Undying from the mouth of her brother Rhaegar. Since the Targaryen house sigil features a three-headed dragon and her ancestors are the ones who united the seven kingdoms, that's even further support that the prophecy is tied to Daenerys.