My family in APO is the "Ashby Tree."
We trace our common fraternal ancestor to a brother with that last name. Although several of our trees have that name, some are named after actual trees, and some are named after hometowns they had in common.
You may want to name yours something that is related to or derived from a symbol of the fraternity. Check out this page
http://www.phimudelta.org/?page=FoundersCreed and pick a word or several words. Maybe you don't want to be "The Justice Family" but maybe "The Justinians" sounds cool.
There are multiple branches on our tree, which helps make large trees smaller. Each branch has some sort of theme or subname.
It's all pretty fun as long as it doesn't become more important than the chapter itself.