In my opinion, it's impossible to be central to "all the tourist sights" in Tokyo. The city is far too large and the sights too widespread to be close to all of them. We stayed in Ueno (near the zoo and several museums) and quite liked it. It seemed a lot of the cheaper options were located there. It was walking distance to the Akihabara and Asakusa districts, but quite a trek to visit Shibuya or Harajuku for instance. The subway in Tokyo is great though. Or at least I thought it was 
Our accommodation in Ueno was modest (New Izu Hotel), but I think it was still over US$100 per night. Actually.. here it is.. 10,500 yen / night - so just over $100. That was a few years ago though. And these were traditional Japanese rooms, so you sleep on a mat on the floor.
On a second visit in Tokyo, we stayed at Hotel Kitcho in the Nihonbashi district. A tiny room, with a tiny private bathroom, but we really liked the area and the staff were very friendly (as most Japanese people are in general). That one was only 9000 yen / night, so would be under $100. Like I say though, a very small room (kind of the norm in Japan mind you). I'd stay there again if visiting Tokyo.
Hope it helps and don't forget to check out the Tokyo guide. It's had a reasonable amount of work done already, but if you have anything to add, feel free to contribute!