There are lots of things you could do with this computer. My first choice, if your mom is happy with it, is to let it just continue to be her PC, changing nothing unless she needs more than it can do as-is. (Well, you might upgrade her hard drives; but otherwise, it's probably adequate for her needs now.)
However, if she has (or wants) a new machine and she's offering this for you to do with as you wish, I have a few ideas for you:
1. Use it as an "experimental/quarantine" computer for trying out new software before you put it on your main computer. If you have never used Linux, I strongly recommend it; if you tried it years ago and gave up because it was too frustrating, try again, it's much more mature and user-friendly today. But don't wipe out your Windows installation; instead, set it up to dual-boot, and keep Windows on it for trying out new Windows apps and opening suspicious e-mail attachments. You can also use it to test questionable hardware when you're trouble-shooting another computer ... as long as the parts are compatible with this machine's motherboard, of course.
2. Convert it to a home server. You can set it up to backup all the other computers on your home network, store and serve media files, etc. You'll want larger hard drives, and you'll need either Windows Home Server or a suitable Linux or BSD that's designed for server use. Once it's setup and running smoothly, you can remove and store the monitor, mouse and keyboard so it takes up less space; hide the "headless" box behind a potted plant if you wish.
3. Set it up in the guest room. Wipe all personal data from the hard drives, provide an Ethernet or Wi-fi connection to the home network, and set up a (non-administrator) guest account for house guests to use. Cousin Marvin will appreciate having a way to check his e-mail and surf the Web when he comes to visit. You will appreciate not having Cousin Marvin use your computer!
4. Give it to Cousin Marvin, or Aunt Winifred, or someone else who can use a good-but-not-cutting-edge computer. Wipe the drives first, of course.
5. If you are in the market for a new computer, either sell it or see if it has any trade-in value.
6. Cannibalize it for parts you can use when something in your main computer fails. You may not want to use that GeForce 8800 in your main machine, but you'll be glad you have it if your high-end video card dies on the Sunday before payday. Store the parts in anti-static bags (I always save them when I buy components) along with the driver disks and manuals. Be sure to label the bags so you can find what you need quickly.
7. Open it up, blow out the dust, then put it (along with all the cables, installation/driver discs, etc.) in a box in your closet. It's always good to have a spare computer available for emergencies. If nothing else, it will let you go online so you get help in the MaximumPC forums when your main computer suddenly dies.