Tokyo has a lot more budget accommodation options than you might realise. While there are lots of interesting and fun hostels and budget hotels, there are other options. It might sound ridiculous, but internet cafes (like KaraNet24) are a cheap, safe and generally clean place to sleep (it’s really OK, they have showers as well as pillows and blankets to make your stay more comfortable!)
If that’s not your thing, there sure are some quirky hostels and tourist hotels. How about too cool for school RetroMetro Backpackers? Or the hoppily cheap Kangaroo Hotel?