If changing the hard disk makes no difference this more or less has to be a motherboard problem. But first check the HD connector on the motherboard, and if it connects to the HD with a ribbon, check that too. Look for any bent, distorted or dirty contacts, and clean them with isopropyl alcohol. That’s not likely to cure it but it’s the only thing you can do easily. Also, check the fan is working and the grill isn’t clogged with dust. Speedfan form almico.com is useful for checking temperatures.
If the problem regularly appears around 5 minutes after switching on it would suggest a faulty solder joint on the motherboard which gets worse as the board warms up and everything expands slightly. Does it fail quicker if the laptop is already warm? Does it run for longer if you first cool it down in the fridge?
If it is indeed a faulty solder joint then the only solution would be to reflow the motherboard. Some people have had success doing this at home but without a proper reflow oven which follows a precise temperature profile it’s likely to be kill-or-cure.