Just a couple of ideas off the top of my head:
Kamakura is a short trip from Tokyo, and features lots of old temples and a giant Buddha. Very historic -- it was the capital about a thousand years ago. I would highly recommend it. For a more aggressive day-trip, you could try to get to Nikko and see the temples there, with the original "see-/hear-/speak-no-evil" monkeys. But if you only have a weekend free, I probably wouldn't try to go that far out.
Within Tokyo, if you are a gadget geek, it could be fun to wander around the electronics shops and stalls in Akihabara and check out the latest cool stuff. You could also try to take in some Kabuki if you have a spare evening (they usually provide a listening device with English explanations with which to listen along while watching). The Imperial Palace grounds and the previously-mentioned Asakusa are also worth short visits. If you're really into history, you could try to hunt up things like the final resting place of the 47 Ronin, for example; there are lots of historical sites scattered here and there around Tokyo.
Tomcat's Oedo Onsen Monogatari recommendation sounds good, too. I haven't been there myself, but would definitely recommend checking out Japanese baths/hot springs. If you wanted to get ambitious, you could take the shinkansen to Hakone and hit some onsens with views of Mt. Fuji -- but again, likely a bit much to pack into your only free weekend.
For contemporary culture, Shibuya and Harajuku are good if you want to see what the kids are into these days, Ginza is for checking out high-priced fashions, and Roppongi has lots of bars where younger foreigners hang out.
When you say you will be "in/near" Tokyo, does that mean you will not be in Tokyo proper? If not, where would you be?
Bpp