What is the minimum voltage that a 5V Pro Mini can take on its RAW pin and still create a stable 5V supply for itself? I looked at the schematic but the voltage regulator type is not specified there...
The 5V version uses a MIC5205 regulator the data sheet says the voltage drop needs to be at least 165mV, so anything over 5.165 volts should do.
Excellent, thanks. 
I have some tiny leftover unregulated 5V power supplies (~7V unloaded), so they should work fine after filtering.