I'd really like to know how to fix this as well. I bought an xbox off eBay. When I got it, I hooked it up to my monitor. I was able to boot to the home screen fine, even play a DVD. However, after about 15 minutes, the screen got scrambled, then went totally black.
I hooked it up to a TV, but the video was still black. The audio was working for a little bit, but that eventually cut out also.
It seems it only works when the console is totally cold. For example, this morning, I turned it on and it displayed the Xbox logo and went to the home screen, but the video cut out only after about 15sec. This tells me there's a problem with the solder connections. Someone had also previously done the x-clamp mod. I have yet to loosen the bolts, as they are pretty tight on there. I get no error messages, and no red lights. I would really like to get this thing up and working.