Yep, it is a fantastic idea to either:
1.) keep your quarantine/hospital filter running in your main tank, or
2.) keep some filter media for the quarantine/hospital in your main tank's filter
Either way works. I do the second option myself. I keep the sponge for my hospital tank's sponge filter in one of the supplemental media compartments of the filter on my main tank. When I need a hospital tank, I wag out the 10-gallon, fill it with (dechlorinated) water, put some gravel in it, put the media in the filter and - viola! - instant cycle. Remember though, that if you have sick fish in the little tank you won't want to put the filter media back in the big tank when you break the little one down again. Throw it away and put a new filter in the main tank to prepare for next time. You don't want to risk introducing whatever sickness the ill fish had into your main community.
Also, I'd encourage you to get a 10-gallon as opposed to a 2.5 or 5-gallon. Those are really just too small for most things to be minimally comfortable in, and a 10-gallon can still be had pretty inexpensively. You'll never regret getting a tank with slightly more space (after you've paid for it, anyway).
pendragon!