You can't rebind the F3 key to something else.

Most keys can be rebound to another key in the Key Binds menu however some keys are missed including the F3 and multi-key combo's (f3-A etc.). 

I don't know if this is is intended or not. It seems like an oversight but I don't know.

Steps to reproduce: 

1. look in the key binds menu

2. don't find F3