Can I power an Arduino with a computer's motherboard?
You want to tap off of the +5VSB pin of the ATX or BTX power supply connector and send it to the 5V pin on the Arduino. Where this will directly drive the VCC(5V) of the UNO. 

 

Note that 5V of the USB basically drives the VCC directly, except through a MOSFET that acts as a blocking diode to prevent local VCC from flowing back to the 5V_USB.



Not sure if you are suggesting that the 5V USB is frying Arduino's. If so that is bad. If true; Fix that first.



That said (with good 5V usb) you should be able to find or configure (BIOS) so that a USB connector will remain powered, and allow a wake up from mouse movement. Hence a possible solution would be to use a Leonardo/Micro's Mouse emulation to do just that as to wake the PC, without any hardware mod's.

Freeduino Micro's are pretty cheap.