[kebisan]

I would suggest Edogawa-ku. Maybe somewhere around Nishi-Kasai or Kasai stations. I lived in that area for a while and loved it. The rent was about 80,000 yen and access to the city was pretty good via the subway system. It will probably be hard to find an apartment if you're only going to be there for 3 months so weekly mansions are definitely the way to go.

Like MMM said, lively and central will definitely mean more expensive and to be honest it's nice to get away from it all sometimes. In my experience, even the residential areas surrounding downtown Tokyo are pretty lively and crowded. Just be sure to choose a place with good access to a big train or subway line. Not only will it be more convenient to get around, but izakaya and karaoke places are always concentrated right outside train stations so you'll always have something to do.