I would say that updating the BIOS should be the last option before changing the hardware as the updating of BIOS is a very risky process and one needs to be very cautious and wary during the process as even a slight mistake can fry your motherboard and the only option left would be to change the entire motherboard.
I don't think there would be a hardware fault, even if there is then most likely it could be either a faulty wire or loose connections. HP tends to use the Seagate hard disks which are indeed extremely good so the chance of that failing would be minimal. (I have HP desktop myself and the disks are pretty robust)
I found the latest driver on Softpedia over here :-
AMD SATA Driver - download tag
Although, if there is an updated version of the driver at the HP website, please download and update that package instead of the above given as sometimes it happens that the system works fine with the OEM drivers only <_<
Please read the instructions carefully before proceeding.
Also, it is not mandatory that the system is not stable if only one BSOD occurs, if you are getting it too frequently then that becomes a problem. So, please perform the above steps and in case of any further problem, please do post back here ^_^