If the system is ok when doing something but locks up when idle, it sounds more like chipset then graphics driver issue.
If you think about it an idle PC wont really use graphics but motherboard will still be doing things.
Win98 isn't exactly stable best of times, but shouldn't be THAT bad.
You can try installing 1 driver, say chipset, test for a while, graphics, test for a while. Basically laptop version of installing 1 piece of hardware at a time.
You can also try using non HP drivers, like what you were suggesting with the graphics card. No doubt a few things will need the official drivers but majority will be ok.
Side note, this was my very 1st laptop from my very first job! Had the Floppy and HDD that fit in the CD drive "multi bay" and wifi adaptor that fits to the top of the laptop.
HDD was actually to boot 2 copies Windows 98, 2k, XP with either Office 2000 or XP for helpdesk troubleshooting.
Nice little thing but my shoulder still hurts/clicks from lugging it round 5 days a week