Well, the Picadilly area will be the most central, and probably the most interesting for a one-nighter. But if you're going for a hotel, this'll be among the most expensive. Granted, the proximity to touristy places, as well as Soho (plenty of cool bars/clubs there) is handy.
It also depends what kind of club/nightlife you're looking for. If you want the megaclub thing, Ministry of Sound is just south of the Thames near the Elephant & Castle tube station, although that's a bit played out. Turnmills is toward the east (Farringdon tube), but last time I tried to show up there without pre-purchasing tix, I was hosed.
If you're into the alternative thing, Camden's got some good live and club venues, as well as the big market area there where you can pick up all kinds of flyers to see what's going on that night. (I was walking around there when I found a flyer for a London night for Velvet - a goth/darkwave club I used to frequent in LA).
For a slightly different vibe (some indie, some laid back grooves) there are places like Notting Hill Arts Club, so you'd head west, to Bayswater/Notting Hill/Kensington.
There are lots of cool/funky late bars and clubs in the Shoreditch area, as well.
You can find listings and reviews of the London nightlife here:
http://viewlondon.co.uk/dance.asp
Overall, where you're staying shouldn't matter. You can quickly catch the tube out to anywhere, and then grab a nightbus (cheap!) or cab (expensive!) back to your lodgings if it's a late one. Some of my favorite London memories come from drunkenly waiting around for and/or riding the nightbus back from the middle of town to Bayswater.