Lot of good suggestions already. I agree with someone who said earlier that it's mostly about having the right mindset... with the right mindset, you'll be able to have a good time in cities that offer far less than Tokyo, let alone Tokyo. Of course, that's easier said than done. I also agree that it'd make the whole thing much easier if you're somehow able to get your friend out of his current mode.
Having said that, some ideas:
- Do a day-trip to Hakone and do onsen-hopping. Tons of great day onsens in Hakone Yumoto. Or you can head up to Yunessen (an onsen theme park) by bus from Hakone Yumoto). If you're in Japan for that long, you've gotta do onsen.
- I don't like shopping either in general, but I could spend a whole day at a place like Shinjuku's Takashimaya Time Square / Tokyu Hands mega-complex, or Kawasaki's La Zona complex (directly adjoined to the station).
- Odaiba and Yokohama's Minatomirai area are both really cool areas to hang out, with lots of food, shopping, giant ferris wheel. Food, shops and ferris wheel do not exactly inspire a lot of excitement in UK or US, but have an open mind for Japan and you'll realize it's a different experience. Minatomirai's ferris wheel runs till 9 or 10p, I believe. You can combine a visit to Minatomirai with a swing over to Yokohama Chinatown. Odaiba is also cool.
- I think Disney Sea (as someone mentioned above) is a great idea.
- How about Tokyo Sky Tree? I've never been, but it's near Asakusa and it sounds like a fun complex and I know it's open at night.
- Sunshine City in Ikebukuro... go to the aquarium, which is awesome with penguins and lots of animals, then go up to the observation deck, then do some shopping/eating and wrap it up with a trip to Gyoza Stadium.
- Go explore random neighborhoods - Ueno (Ameyokocho), Kichijoji (cool neighborhood north of station, plus Inokashira Park), Shimokitazawa.
- Tokyo Station is an attraction in and of itself. Restaurants, bento shops, ramen, anime character shops, so many cool things... and it stays busy into the night.