My first guess would be the power supply. I use a
FrozenCPU Ultimate PSU Tester for power supply (PSU) testing when I am away from home. It is not as good as a qualified technician testing the PSU under a
"true" (realistic) load with an oscilloscope or power analyzer, but close. The advantage of this model is that it has an LCD readout of the voltage. With an actual voltage readout, you can better detect a "failing" PSU, or one barely within specified tolerances. Lesser models use LEDs to indicate the voltage is just within some "range". These are less informative, considerably cheaper, but still useful for detecting PSUs that have already "failed".
Newegg has several testers to choose from. All these testers contain a "dummy load" to fool the PSU into thinking it is connected to a motherboard, and therefore allows the PSU to power on, if able, without being attached to a motherboard - great for testing fans, but again, not a true load. Alternatively, you can swap out the PSU with a known good one that meets the computer's power requirements.
I do not recommend using a multimeter to test power supplies. To do it properly, that is, under a realistic load, the voltages on all the pins must be measured while the PSU is attached to the motherboard and the computer powered on. This then requires poking (with some considerable force) two, hard and sharp, highly conductive, meter probes into the heart of the computer. One tiny slip can destroy the motherboard, and everything plugged into it.
