Powering an Arduino Pro Mini directly from a LiPo
The only thing about the 3.3V Pro Mini that is actually 3.3V is the 3.3V regulator.  The main chip (pretty much the only other component) is good for anything from 1.8V up to 5.5V.

Running direct from a 3.7V LiPo or Li-Ion will be perfectly fine.  You might want to disconnect or otherwise disable the regulator though to reduce current consumption - the latest Minis have a solder jumper SJ1 for doing just that, but older versions you may want to actually cut the 3.3V OUT pin of the regulator or de-solder it entirely.