What does one do the summer between graduation and graduate school? I am about to graduate with a B.S. in mathematics and will (hopefully) be attending a PhD program somewhere starting in the fall.

What is one supposed to do the summer between?  I don't have any money, so I can't "travel."  I would love to do another REU, but I don't think they generally admit graduates.
As someone who jumped keenly straight from my undergraduate into graduate school (we could start at any time), my advice is to rest. Even if you cannot travel, spent time with your parents or grand parents or anyone who will give you a room to sleep in. Maybe help out at a local charity. Clear your mind. Read some novels. Go hiking. 

This could be the last extended break of your life (before retirement). Enjoy it.