I've bought a cheap broken laptop on eBay for around 20EUR with the description "turns on briefly the fan and the shuts down, no image".
The laptop was actually in pretty good condition, came with a working battery and charger and RAM, but without any hard-drive.
A quick search on the Internet turned up the fact that these laptops usually die from heat, having inadequate cooling on the integrated video card.
I recovered the laptop by using a makeshift heat shield made from a sandwich of aluminium foil and cardboard an using a heatgun (paint stripper) to slowly heat the chip. A blob of leaded solder was used as reflow indicator, knowing that the leadfree solder melts at higher temperatures and the BGA balls take a while to heat up (being under the chip).
A cheap multimeter was used as a temperature indicator, with the thermocouple pressed against the chip body.
The cardboard started smoking after a while but this did not affect the operation in any way. The actual heating process took around 2 minutes, another 30-40 seconds were used to slowly pull the heat gun away from the board and it was left for ambient cooling another 10-15 minutes.
I won't bore you with stories, I think the pictures below tell all the details.
The thermocouple shows only 16 degrees because I had to open the balcony door to vent the room since the smoke sensor started beeping after the cardboard started smoking.
This "operation" was done about one year ago when I had no proper tools to do this. Now I have both a hot air station and an IR thermometer but haven't had the chance to use them on something similar.
The laptop was used for ~6 months by my father but then it died again of the same cause. Being used on the bed with no air flow might have had something to do with it.
I might attempt a reball just to see how it turns out.