Game Of Thrones: Is Tyrion Lannister Actually A Targaryen?
The Dragon Has Three Heads
This is the prophecy that Daenerys hears in the House of the Undying, and a major catalyst for both this theory and R+L=J, with Tyrion, Dany and Jon being the three heads of the dragon.
We know Daenerys has three dragons, so it would fit that there would need to be three people able to ride them.
Each one of them killed their mother during childbirth: we now know for sure that Lyanna died giving birth to Jon, while Joanna and Rhaella died when birthing Tyrion and Daenerys respectively. They've all had to kill someone they loved: Jon killed Ygritte (or at least feels responsible for her death); Tyrion killed Shae; Daenerys killed Khal Drogo.
It fits in with the prophecy, which would lead to them riding the dragons to save the Seven Kingdoms, and gives further explanation as to why the three characters have been established as the leads in both books and show.