I remember a list of Tokyo economy hotels in that area. An American traveler said she liked the Aizuya Inn. The Kangaroo Hotel looked pretty nice for a budget price range. Tiny rooms, but hipster feel. Haven't stayed there myself, though.

The YMCA Asia Youth Center Hotel is one option. The location seems to have changed; now it's close to the Imperial Palace. Back when I went years ago, it was closer to Aoyama-Itchome metro station.

If you're interested in hostels, Quality Hostel K's House Tokyo Oasis was an excellent hostel, one of the nicest I've stayed in while traveling around Asia. Felt traditional, but with all the modern amenities. They have another branch in Tokyo that feels more like a standard backpacker hostel, clean but functional.

I've actually made a video of my best Japan budget travel tips. Not sure if I'm allowed to put a link to my website here. So I just sent you a private message.