Check the caps if they have started to leak. They may be bowed, bulging, or have soldered connections that look dull or dull green.
Check the PSU, unplug it from the motherboard and connect it to a hdd to give it a 12 V load and start it and measure the power output on the connector, or carefully check voltages while still connected to give it a load on all pins.
Those are reasonable suggestions, but the battery is a good bet too; if it is not holding a charge, then it is dead and removing it will let you see damage underneath. Also reseat all the socketed chips, as heat and cold cause expansion/contraction and will mess up connections. Lightly sanding the pins on the chips will remove oxidation; remove any Zorro boards and de-oxidize them with a rubber eraser on a pencil -- cleanse with IPA afterwards. A toothbrush scrubing of the Zorro sockets and CPU connector will help there.
