I like the name
Alexandria. I do not think parents should take nicknaming matters into their own hands. Just because you despise the nickname
Alex, doesn't mean she would (and you may like
Ally or Andi, and she may not). I think nicknames should be left up to the child, not planned by the parent(s). Parents choose the formal name, so let the child choose his/her nickname(s) (if they would even want one).