A friend's desktop computer (random clone, motherboard Gigabyte Pentium 4 Titan 533 Series GA-8IGX with an Intel 845G chipset, assembled in 2002), stopped working recently. Only a light on the motherboard (the manual identifies a 'DIMM LED') and the power light on the Ethernet card come on. He replaced the power supply (HEC-300AR-T) in 2009. He replaced it again this time but the new PS made no difference.

I suggested that the motherboard has failed. Has anyone other ideas?

I already recommended that he replace his computer.

bad RAM? it would be worth swapping it out and seeing if that makes a difference.

The first thing I would do is check all connections, one at a time, unplug and reinsert them (i.e. clean the contacts).
(Just because the mobo is from 2002 doesn't mean its useless.)
Check that the ram modules are seated correctly, remove and reinsert, (also the harddrive terminals).
Still not working, look for 'blown' capacitors, (domed heads instead of being flat), this would mean a new mobo.
Edit: Check the bios button battery.

Also, have a look at the sticky post at the beginning of this section which covers do-it-yourself diagnosis. If the PSU fan is running but the computer isn't even attempting to boot, it could be a dead motherboard, CPU, or HD.

I recommended this. The symptoms aren't those of bad RAM.

Done.

Otherwise this exercise would have no point.

The CMOS battery? Not-booting wouldn't be a symptom.
This motherboard has 2 BIOSes, a back-up in case the primary fails.

Already done. I had hoped that responders would have assumed I had done all the standard stuff. I was hoping there might be something unusual about this motherboard that someone might know.

Do the fans come on ? Any beeps ? If the fans don't work, then it's probably the mobo.

So the PSU was replaced because it was known bad, or in an attempt to fix some symptom ? If the PSU was replaced and the symptoms persistent then it is not the PSU. That leaves the mobo and CPU, assuming you've already ruled out RAM like said above even tho it is unlikely.

The only option now is to try new parts and see if it solves the problem, e.g. a spare CPU and mobo if you have some to try. It is very difficult to differentiate between the two if the system won't boot. If I had to bet I would bet on the mobo, because the CPU tends to fail more gracefully and the fans would be running.