Since I love spending someone else's money, I vote for the new mini!
If you have a monitor it is a tough to beat and the performance increase from your eMac to a Core 2 Duo will amaze you.
Also, you get a GB of RAM (should be fine for your uses) and an 80GB hard drive which I bet is substantially larger than you currently have.
Plus, to make it easier to swallow, Leopard and iLife '08 are included and that's over $200 worth of software.
If you don't happen to have a display, or if the one you have has a lot of miles on it, I would suggest taking a look at the new iMacs.
(I could be biased, here
For about the price of a good 20" display, speakers, keyboard and mouse, iMac gives you that plus a larger, faster HD (250GB), a more powerful graphics card, built-in iSight camera and built-in stereo speakers all in a neat space-saving package. And don't forget the same OS + software mentioned above AND room to upgrade to 4GB RAM! AND a faster frontside bus (800 MHz FSB compared to 667MHz FSB on the mini!)
Making this decision even harder.
I like spending other people's money too.
With above options you could start using it for work, too. Macs are quite friendly with PCs now.
OTOH: Least painful on wallet option:
If you backup any important files you want to keep (archive and install), then format the HD with the write zero to the disk option checked then reinstall OS X. You would save a bundle o' cash and could use it till it falls apart.
Next to the least painful option:
Buy a new HD (internal or external) install OS X on new HD, migrate apps and files to new HD, zero out old HD and use for backup. Then nickel & dime it when things start to fall apart. Who knows? Maybe you could use it for another 4 years!