Historically, Dragonstone served as the seat for the Crown Prince of the Targaryen Royal Family. So to speak, the heir apparent to ascend the Iron Throne next. With the events of the Game of Thrones Books, Dragonstone fell to the Baratheons. But, when Stannis left the island for the North, it was taken over by the Lannisters/Tyrells who supported Joffrey. Who the island would have reverted back to is something of a mystery. I would assume that Danaerys and Aegon Targaryen would have reclaimed their old ancestral home, keeping it for a rainy day. The island, if it were under Targaryen control at the time of the Aegon X's death, would have been home to Daenys Targaryen, as he had not been officially removed from the heir apparent spot until after the king's death. 

It could very well be jointly controlled by Baratheon and Targaryen forces. Bluetommy would be greatly appreciated in clearing this up.

If you are in charge of da big bad island, then I'd be thrilled to collab with you.