Are the mini's attached to a UPS?
If so, does the problem persist when the USB<>Data line is disconnected?
Can you catch WHEN at night (perhaps use LogLeech to determine shutdown times)?
Do you know if they are proper shutdowns or sudden loss of power shutdowns?
New cleaning lady using the outlet for her vacuum? (only partially kidding here).
When I first read the title I thought "heat problem" but 3 mini's all at the same time... hmmm.... unless they are all in a really hot room together.
Check ventilation.
If you check "automatically startup after a power failure" in power settings, any change?
How are the monitors told to power cycle? Could they be accidentally powering off other devices attached to the same circuit? (I'm thinking of a programmable power outlet)...
Are there any other devices also losing power? Staff reporting, "yeah, and the microwave keeps resetting to 12:00"... that sort of thing which suggests a breaker blowing... or a switched wall socket.
Just some thoughts.