MacBook screen doesn't turn on My old, black (irony), pre-unibody (2009?) MacBook won't properly turn on. The screen just stays black. I can't hear the startup sound and the keyboard caps lock light won't shine, either. The power light, hard drive, and fans both start and work.

I tried connecting it to TV (via mini-DVI to HDMI adapter) and I can't see anything, although TV thinks that there is some signal coming.

It didn't happen suddenly - previously it sometimes started, sometimes not. Once working it would work for weeks until switched off manually. Then one day it wouldn't start at all.

I think that the motherboard could be near death, but the fact that some components work gives me hope.

Any suggestions are welcome. I have a new Mac now and all data has been backed up so I can try pretty much anything.
Try resetting the PRAM and SMC. 

How to reset PRAM: restart or power up your Mac and immediately hold down the Command ⌘+Option ⌥+P+R keys all at the same time. Keep holding them down until the Mac resets. Also, on these old models you used to have to do it a few times: i.e., just keep holding down and let it reset a few times, before it would be completely clean.

Resetting the SMC is a bit more model-specific and I can't remember off the top of my head what you needed to do with your model, but you'll find instructions here, or you can Google your model number with "reset SMC".