What I used to do with IBM Thinkpads T41 model that had a problem with the ATI video breaking the solder connections was lightly press down on the video chip and then power on the laptop and if it worked that was where the problem. Thinkpad maintenance manuals are available to all. Usually HP & Dell manuals too. Only go to the maker not one of these manual sites that want money or worse.
There was another Compaq NC6000 laptop that pressing on the right area of the keyboard as a test and then powering on told where the problem was. Using a fine soldering iron and resoldering all the lines on the Max chip fixed that one. But not a later model.
Trying a reflow may work but usually only for a short time. You'll probably see that suggested somewhere.
On the dv9000 HP for example there were people that swore adding a copper shim between the video chip and the heatsink that added more cooling surface was the key, but only if done before it failed.
If you are feeling super brave and do not care if you damage the laptop beyond repair then the HP maint manual could help you narrow it down. I say beyond repair as the connectors for the speakers, touchpad, keyboard, power switch, indicator board lights and sometimes daughtercards for USB ports are very delicate and easy to damage.
If you decide to open it up post pictures of the connectors and I can help on the proper method. Some flip up, some pull a little clip, some just poll out and so on. Very easy to damage. I liken the proper strength to using your two pinky fingers.
Don't even think about used motherboards if it is a video chip issue. They'll fail. HPs first cure was a Bios update that ran the fan stronger for better cooling.
This post has been edited by rotor123: 28 December 2011 - 10:53 AM