Well, I'd recommend you experience an earthquake if you can: that's a pretty unique experience, just try to avoid the strong ones!
Seriously though, if you have 3 days you have plenty of time to head down to the Izu peninsula to try one of the authentic Japanese Onsen (Hot Spa's). My favourite is here: http://journals.worldnomads.com/simon_monk/story/13656/Japan/Indelible-impressions-from-Japanese-Hot-Springs
Another experience I'd recommend would be to get up early and take the train to Kamakura for the temples and food too. It makes for a wonderful day out: http://www.japan-guide.com/e/e2166.html about 10 years ago
Part of this depends on your travel dates. Do you know when you'll be here? I'd like to recommend a local festival or cultural event if there is one on during your stay... but need to know your dates first.
Besides that, here's a list of ideas to get you started:
1. Take a stroll round Asakusa - the old part of Tokyo and enjoy Sensoji Temple and Shrine.
2. Find a less central shopping area like Musashi Koyama and join the locals on Saturday or Sunday. It's a lot of fun and you'll also find some great bargains compared to shopping in the more 'touristy' areas.
3. Visit the Imperial Palace and the East gardens for a little bit of history. Right next door is the super funky Marunouchi district for super high end shopping and restaurants. Nice bit of contrast for you : )
4. Odaiba is the fun zone with a future science museum (the Mirai museum) with some great interactive displays and challenges for kids. Then there's Joypolis, an indoor games center, and Toyota Megaweb if any of your family are into cars. Then head over to Venus Fort for some indoor/outdoor shopping. This mall is famous for being designed inside as if you're actually outside : )
5. Spend 1 night at a ryokan with an onsen (hot spring) at a nearby district like Kamakura, Hakone, or Karuizawa. Very Japanese and very unique. Love it.
That should get you started Liz. Hope it helps : ) about 10 years ago
Asakusa: as Honor Dargan stated above. But I would also mention Asakusa markets is the place to get all your gifts/novelties. I would also recommend going to the Kamiya Bar. Its best in the evenings and it gets really busy, I went there a few times by myself and just had a great time meeting real Japanese ppl. Its a very old bar in Tokyo.
Tsukiji: Visit the Tokyo fish markets and then find some really fresh seafood at one of the little shops. I really enjoyed doing that. But you have to get there really early.
Ikebukuro: I went to a cat petting zoo called Nekobukuro. 600Y. Its just downright bizarre, thats why I think its worth a look.
Harajuku: If you can get to Harajuku on the weekend and go to the street mall you might be lucky to see all the girls dressed up in lolita style.
I went on a cruise of the river at night and that was pretty fun, it included meal and unlimited drinks (sake, beer, wine). Try and find some Nigori Sake when u are over there it was favourite type its not like normal sake.
I used the Lonely Planet Phrase book and found it came in handy alot. Its actually really easy to get around the train stations have the sings in english and japanese. The Yamanote line is JR and I based most of my week around that line and if you think it is cheaper you should buy daily tickets. But be aware there is 2 private train lines and then the JR line. and you have to buy a ticket for each train line. But its easy peasy. about 10 years ago