Could overheating cause this problem?
Overheating can cause a lot of problems, in fact, I'm sitting hare with an HP now, the CPU & HDD runs, but it won't boot, no BIOS screen, no keyboard control. Have replaced the PSU to no avail. It could be a portion of your MB damaged that controls the notebook screen, but the external output was spared.
Now I have to decide, to replace the MB and/or the inverter you mention above, or sell the PC for parts. DDR2 RAM is pricey, so that's something.
And yes, mine was overheating, but because it was an AIO unit, was afraid to damage it further be disassembly. Once it stopped working, had to.
So you replaced the HDD too. Did you reinstall Windows? This is kind of puzzling, as you have a external monitor that runs fine, but the main one doesn't. And the computer has Windows Vista. So given it's age, I'd say no to sinking a ton more cash in, but if a simple fix, yes. It's not worth a MB replacement, no way could you recoup your money & maybe can't now.
Check this out, pertains to cables, you've changed the LCD, see about the rest. Note that cables does go bad, a hot environment doesn't help.