I can recommend a great hostel in the Asakusa area, right behind Sensoji Temple (where Lamia also recommended the Chisun Inn): the Sakura Hostel. I have stayed there on numerous occasions, and can’t recommend it highly enough: the facilities are immaculate, the staff is fantastic (lots of young English-speaking people who will go to great lengths to help you out) and the neighborhood can’t be beat, plus you can book online. It’s a great place to meet people of all ages, also. When I’m in the Tokyo area without my husband, I’ll usually save money by getting a dorm bed there (and they do have female-only dorm rooms). The same company also has similar hotels, etc. around Tokyo (they’re linked from that website).