What I usually do is play "strip it down". You have a system that doesn't work, so just keep stripping it down on it's component parts (while still allowing it to boot up, of course), until you find the troublesome component.
Before any of this though, with a good flashlight, double check all the jumpers and connections. Those little jumpers can easily be put back so they're on *one* of the pins they should be on, (and look OK), but be entirely off the second pin that they should be on. Connections can also work loose, so check 'em all. Bent pins under a connector can't be seen without disconnecting the connector, and really checking carefully.
Dirt or hair, spiders, ant's been inside? Temps and voltages look OK? No damaged or sharply bent cables or wires? No damaged traces or cracked/loose solder connections on the board? Capacitors, etc., all look good?
If it has a hd, disconnect the hd, and boot it from the optical drive.
If it has 2 sticks of ram, boot it with just one, and swap the slots that are used if it's still a problem.
If it won't boot up into the OS, see if it will continue to run for a long time, just at the BIOS screen.
Any parts that you can swap around that are known to be good and compatible, are very helpful. You may never find the bad component without substituting several parts to help further isolate the problem.
Good luck!
