If you're thinking of hacking the cam anyway, take it apart and check the tightness of the ribbon cable that attaches to the LCD screen. Also trace that ribbon back to it's origin and check that side of the connection as well. It may help to actually remove the ribbon and clean the contacts with alcohol. To remove the ribbon, flip up the little plastic bar where it ties into the screen, then it should pull out. Now for the crazy part I mentioned earlier...
It's a pretty commonly known trick among Sony homebrewers that you can many times get by with rapping the front of the cam with your knuckles for LCD issues. Apply the blow right where the Cybershot logo is located. Turn cam off prior, then give it a good hard rap, then turn back on to test. I fixed one that way this spring and I think it took off and ran on the 4th attempt. It's still in the woods and going strong. Sounds rediculous but it's worked for many folks, myself included. Often times this is used to fix a WHITE screen though, I've heard less about black screens.
If all that fails, as a last resort you could try to temporarily jumper the fuse for the LCD with a pair of tweezers or similar and see if that springs it to life. We can cross that bridge if it comes to it, hopefully it won't be that involved. I listed these options because I'm afraid that Sony won't tell you what you really need to know to fix it and for them to fix it would likely cost more than the cam is worth.
