It's ~probably~ a short on one of the boards or monitor, I would suggest it's one of the auxilliary voltages (12, -12, etc) and possibly a tantalum capacitor
-looking at the faults on my machines, and others reported here it seems to be really common on machines that age.
If you have a spare AT power supply (old type) the compaq boards are pretty standard, so it's pretty easy to narrow the fault down by substituting boards.
The only major problem is getting a Torx screwdriver the right size, and sufficient patience to take the inner case apart!
If it's got a hard disk in, and you've got storage space, it's probably worth $15 for parts, the portable IIs come up on ebay about once every 3 months. make sure you don't get stung on the postage, it's not light!