Although I haven't stayed at them, I have heard good things about K's House (which I think jib71 linked to) and Azabu Juban International Guess T House.

You can also check Sakura House. Or, if you're really looking to crash with people, you can CouchSurf.

What would make it easier is if you knew of an area that you wanted to base your trip out of - Guest T House would give you a good base for the Azabu/Hiroo/Roppongi area; K's House would put you squarely in Asakusa/Ueno area. Or, of course, if you could tell us what the purpose of the trip is - sightseeing? going out? adventure?