Also, if you guys want to get super nerdy, some surname choices could be: Azuma (which literally means 'East') for Azure Dragon, or Minami (Which literally means 'South') for Vermilion Bird. Common surnames with 'Nishi' (for West) include "Nishiyama" or "Nishino". I don't know what surnames have 'North' in them though.
EDIT: Other seasons include Haru for Spring, Natsu for Summer, and Fuyu for Winter. Haruko, Natsuko, and Fuyuko are all actual names, though it might get a little too cheesy if we followed that convention.