Quite honestly I'm not sure what setting the BIOS date would have to do with anything, I'm not aware of any restore disks that require the computer to think it is any special date and that is the only reason I can think you would need to do such a thing.
Could you answer a few questions please ...
Was your hard drive replaced? I have heard some reports where the manufacturer must have done some tweaks regarding the hard drive to make it "theirs" and then when the hard drive has to be replaced for any reason there were issues using the restore media. If that could be the case then HP would have to tell you how to handle this new hard drive; usually they want to sell you one from them but that is just ridiculous and if pushed I'm sure they could tell you what you need to do.
Also, have you backed up all your data? Because a restore to factory settings should wipe out all your data.
Do you know your W7 Home Premium product key (I am only looking for a yes or no, please do not post your product key online ... ever!)? It may be on a sticker on the computer. Also, is this a laptop? Some laptops use SLIC technology and if so then the Product Key should be stored in the BIOS.
So your computer is still running Ultimate even after you tried to restore and it ended up incomplete? It must not have really done much of anything ... or did you son put Ultimate back on after the restore failure?
You say things don't run right ... could you be more specific? Is windows waiting several minutes to boot up? Is the desktop background black with a Not Genuine message in the bottom right corner? These are signs that your son had a trial or invalid key. But if it is something else ... other programs not working, for example, then if you could provide more details about that it might point to other issues, possibly hardware failure. You mentioned it says healthy but have you run a diskscan ... if you select the drive and right-click on properties, then chose the Tools Tab and "check the drive for errors"?