I have 3 HP PCs:
Vectra XA 5/166
I’ve added hard drives to all of them over the years, and none of them has ever complained. They use Phoenix BIOSes, which are admittedly less configurable than AMI/Award, but they all accept new disk drives readily. Just make sure jumpers are set correctly and go into the BIOS on first boot to set it up right. Watch out - OEMs like cable select!
I’ve used the same sort of tools Dee-ehn is talking about, though quite a while ago. Worked fine. It claimed to be a BIOS addon, but I’m pretty sure it just put some special boot files on the hard drive.
If there’s only room for 1 hard drive (and on many Pavilions this could actually be a problem), use a program like Norton Ghost to clone from old hard drive to the new one.