Melina Tashir
Aes Sedai
I had never thought about the fact that Galad had a claim to the throne... But he would have one, and stronger than Elayne. Why was that never brought up?
To be honest, I don't think Rand thought much about Galad or Gawyn after meeting them that first time. Maybe he did a bit after he deduced who his mother was. But he never had much plans for the other two. As for why Elayne, I suppose because he was in love with her :waggle. What better way to say I love you than to give a kingdom to someone