Modifications
By performing a custom hardware modification on your Xbox, you can transform it and its capabilities. The extent to which you modify your hardware is up to you---and largely dependent on how tech-savvy you are.
You may wish to perform a serious hack job that requires purchasing a special chip online, cracking open your Xbox and getting to work with a soldering iron. This may be intimidating, however, and getting it wrong will almost certainly junk your machine---fortunately, they aren't expensive to replace.
If you prefer, you could perform a "soft" modification that reformats the operating system without making any physical alterations.
The benefit to these modifications is that they grant you freedom in how you use your console---they can allow you to back up game discs or even use the Xbox as a digital video recorder for your television. The drawback is that it is risky and voids any warranty you may still have.
Collect
Perhaps the most obvious solution to using an old Xbox is to play games on it. Though that may sound pedestrian, this console should not be written off simply because it is old and less powerful than modern systems. The Xbox is home to modern classics like "Halo" and "Star Wars: Knights of the Old Republic." These games and others, many of which have sold more than one million copies since release, are a testament to the system's popularity and the quality of its games.
Since the console is relatively antiquated compared to modern gaming marvels, its games are almost universally inexpensive--you can build a respectable library of games that will keep you busy and happy for a long time with very little cash.
Case Mods
Since old Xboxes are relatively inexpensive, you may not be concerned about accidentally destroying it. Should that be the case, experimenting with a case mod may be a way to get some extra mileage out of your system. A case mod is a modification performed on the body of the hardware itself--either the interior workings, the case that contains them or both--for the sake of novelty and experimentation.
There are two ways you can experiment. The first is to remove the guts of the system and put them in something else, like a Super Nintendo. The other option is to gut another piece of hardware, like a Blu-Ray player or a VCR--and install the circuitry into the Xbox case. Modifications like these range from professional and sleek to shabby with a DIY-kitsch appeal--it depends both on the aesthetic you want and the limits of your capabilities.