Different sizes would be good - maybe a "mini game" with 2-3 pieces, or a "mega game" with 5 pieces.

Placing your own pieces is a cool idea - maybe do it "blindly" so you don't know where the opponents pieces are before you start - possible giving you a slight advantage/disadvantage.

Or my favorite idea of all (Same Idea I have for Battleboats!) - play the game on a "hex" type board. Of course on Battleboats, that give you an extra direction to place your piece, in Amazons that would actually take away 2 directions to move a piece.