You could do it pretty cheap, but I would not reuse a 1.6 GB harddrive. It's just too slow and too small for a new system.
You could reuse the floppy, cd drive (though the cd-drive is going to be slow, but that's not really too important), mouse, keyboard (may require a $4 adapter, or better yet buy a $10 keyboard).
Although that system above would work, you would be MUCH better off in the long run if you could spend about $100 more.
For example: According to pricewatch (
http://www.pricewatch.com), monarch computers has a combo deal with an AMD Duron 1 Ghz processor, MSI K7N420 Pro (based on nForce chipset), and AMD approved heatsink for $235.
You are going to need a case (w 300W powersupply) - $40 <> $50.
RAM - (this board is going to require DDR ram) 128 Mb of that would cost around $40 (maybe a little less). Look at
http://www.crucial.com for PC 2100 DDR SDRAM and at the 128 Mb size.
Realistically you are also going to need a harddrive. I'd try to find at least a 15 GB model, which is going to run you around $60.
Since the nForce board has integrated GF 2 MX 400 video and Soundblaster Live Platinum quality sound you no longer need those two components. Top that off with the fact that the K7N420 Pro will support (so far) up to an Athlon XP 2000+ CPU and you have one heck of an upgradable system there. (Yes, the board still has an AGP slot for upgrading the video should you choose to do so.)
You could save a little bit of money by dropping to a lower speed CPU, but I strongly suggest getting the 1 Gig Duron, as it added the SSE instruction set and is only $15 dollars more than an 850 Mhz.
One last thing that no one has mentioned: What about the operating system? That HP is (if you are lucky) going to have Windows 95. Windows 95 is just too old to work with a newer system. Not only is it unstable, but there is very little driver support and Win95 does not support USB. Windows 98 or higher is really needed. (Maybe you've got a Windows 98 CD, I don't know.)
If you add everything up, not including the operating system, you have around $360 - $390 + shipping. Spending that extra $100 is gaining you a lot more speed and upgradability. Maybe Grandma would throw in the extra...

I would not go used, for that amount of money you are not going to get anything worthwhile.