All right, after looking over these 6 pages and Googling this issue endlessly, I've got to chime in here on a topic nobody seems to be addressing here.
This is a design flaw and HP needs to issue a recall.
Look at the replies in this thread. How many people have come here saying their laptop does EXACTLY the same thing the OP mentioned? This should never happen to a laptop that's designed with appropriate thermal management and manufacturing practices. The solder joints on the northbridge fail like *EVERY* other AMD-based laptop I see come through our shop with a no-video problem. Intel laptops never exhibit this same behavior - they sometimes need a RAM reseating or embedded controller "hard reset" (remove battery, hold power 20-30 seconds). But they never come back after that. Only the AMDs end up being a case of "sorry, you should look at buying a new computer and chalk this up as a bad purchase". Typically HPs, too.
The laptop we have in the shop right now is an HP Pavilion dv7-4051nr, product number WQ870UA#ABA. It came in to the shop for no display on startup ("monitor doesn't work" as described on ticket... but technically a "no-POST", but you can see how even basic check-in techs don't know how to identify a no-POST from a faulty LCD). I originally managed to get the PC running again by resetting the EC - pulling both batteries and holding the power button seemed to do the trick. It was also updated with the latest BIOS after that.
But it came back with a further-degraded case of the same problem not even a week later. Now, it won't come up at all - reset doesn't fix it, reseating RAM, replacing RAM, holding/hitting F8, any of the tricks mentioned so far. If I called HP or had the customer call HP, I guarantee they'd state the motherboard is defective.
As is everyone else's laptop motherboard in this thread.
I've seen it enough times to know exactly what happens - poorly-laid solder joints either inside the FC-BGA northbridge chip package, or between the chip package and the motherboard. Heat from the chip isn't dissipated properly, or is allowed to build up too high before the fan kicks up its speed, and that heating/cooling causes the contacts to break. The contacts may squeeze back together for brief periods, making it seem to be "fixed" for short moments, but there's no reliable fix to get those contacts back together (at least, without very delicate work with a hot-air reflow setup, and a whole new AMD northbridge chip).
The issue first occurred in HP dv2000/6000/9000 laptops, circa 2005-2007. My big question is... why hasn't HP learned from these failures - which they were forced to issue a high-profile recall program for - and designed better cooling in these AMD laptops? (the affected dv2/6/9000 PCs were AMD-based only, and nVidia provided the northbridge with integrated GPU just like the AMD chipsets of today) Or why hasn't AMD learned to make FC-BGA packages that don't break so easily? Intel seems to have figured that out before the problem even occurred - I *VERY* rarely see any Intel-based laptops with motherboard failures. So few, I struggle to even think of a single one (that's chipset-related, not abuse/damage-related). AMDs come in multiple times a month with this same issue, and we have to give them the horrible news that it'd be a $300+ repair operation on a $300 computer.
So, the big question for right now is, what is HP going to do about this - if anything?