Done! I put a hole big enough for a #4 machine screw into two places. One as @@RobG mentioned -- it went through the TI logo on the back side. The other was through the rocketship. Board still works!
Thanks also for your comment, @@jpnorair. That's basically what I did, except I was eyeballing them. My biggest fear was that this was a 4-layer board and I'd punch through something important. I'm somewhat surprised by the simplicity of the board, based on just those two layers.