Like others have said they are memory storage devices. I believe Rowena is collecting them because she is trying to unearth the secrets to the Allagan race that came before us. My theory is that when we are done with Eorzea the story will "uncover" some sort of hidden land no one knew about that the Allagans populated. Either that or Square could do another time-travel expansion like Wings of the Goddess in 11 and the tomestones could have given us the key to being able to travel back in time and see the world before Dalamud was launched into orbit.