I had a Cube for over 10 years. In fact, I just sold it on eBay. I had run everything from Mac OS 9.1 through Mac OS X 10.5.8 on it.
Any of the last three flavors it is able to use (10.3, 10.4, 10.5) will work. You'll need an 800mhz third-party processor upgrade or install to the Cube from another PPC Mac with that spec or better via FireWire Target Disk Mode in order to get 10.5 on it.
10.3 (latest revision 10.3.9) is very, very fast on the Cube. The only real downside is the lack of a good modern browser that will run under 10.3. Lots of newer PPC software won't run on 10.3, either. That may or may not be an issue for you, depending on what use you plan to put your Cube.
10.4 (latest revision 10.4.11) won't be as fast as 10.3 but is the latest major release that will allow you to run the Classic environment. Still can't run the more recent Safari releases or other good modern browsers. The ability to run the Classic environment isn't really a big advantage in my opinion since the Cube can boot into Mac OS 9 without even requiring a separate partition for it.
10.5 (latest revision 10.5.8) will run about as fast as 10.4. Apple is still releasing security updates for it, and it will run recent Safari releases (and some other modern-ish browsers) which means that no web doors will be closed to you. The only disadvantage is that you can't run the Classic environment, but again, the Cube can boot into Mac OS 9 so this shouldn't be a significant issue.
So... I'd recommend keeping your Mac OS 9 installation and installing 10.5 over it. If you can. Only if you feel that 10.5 runs too slowly on your Cube (only you can decide that for yourself, since it's very subjective and depends on too many factors to go into here), wipe it and try 10.3.