Try the battery on the motherboard.
This sounds very much like this is the problem. When you switch on from an overnight 'off', the battery is dead. Being some sort of rechargable battery, when you first turn it on it picks up a little charge, then works fine the rest of the day. Then dies again overnight.
The computer won't finish the post because the BIOS has been reset to factory, reconfigures itself on first boot (but doesn't complete the post because of all the 'new' devices it detects), then when you reboot it works fine (because the little charge the battery got is enough to store the 'changes' in the short time it takes you to reboot).
Now for the bad news - being an HP, the battery is probably hard soldered onto the motherboard. All is not lost however, there should be a jumper switch to configure for an external battery, you will have to consult your manuals (or HP's website) for the specs. Turn the PC off, find and configure the jumper, plug in the external battery (assuming you went out and bought one - make sure you get the right one as per HP's specs), then boot as normal (you may still have the same problem on the very first boot but that should be it)
If you are lucky and its not hard soldered on then take the battery into your local battery store and buy a new one.
My System Specs:
Cameras: 2 x Canon XL1s
Computer: Dual Intel Xeon 2.8Ghz processors (hyperthreaded) - Intel SE7505VB2 mother board - 2 Gb PC2100 RAM kingston -Adaptec FireConnect 4300 -2xseagate 10K scsi hdd 150G ea -2xseagate 7200 ide hdd 200G ea - Plextor PX-708A DVD burner - ATi Raedon 9800Pro video with Dual SUN 21\" monitors - Sound Blaster Platinum live!
Software: Windows XP Pro with SP2 - Pyro Pro (adobe\'s premier pro, encore DVD, and audition bundle)