OK, so it sounds like it is thermally related, but not necessarily overheating. Probably a loose connection, bad solder joint, or cracked PCB trace, where thermal expansion/contraction due to temperature changes causes the connection to get better or worse.
If you can run the unit with the PCB exposed, try tapping on components and connectors with a non-conductive (plastic or wood) object to try and isolate the problem area. Be careful, there are high voltages in the backlight inverter!
Assuming you've identified the problem area, try re-seating any nearby connectors or cables as a first step. A bad solder joint may not be practical as a DIY repair if there are a lot of surface mount (especially BGA) components, and if it is a cracked PCB trace you are probably hosed unless the damage is obvious.
If you want to try replacing boards, I have no idea where you would get replacements.
Nostalgia isn't what it used to be.