Some years ago I backpacked through Japan. It is still one of the highlights of my traveling career. We are now considering going to Japan with (small) children. Based on my earlier trip I consider Kyoto to be a nice focal point for such a trip. Back then, I travelled a lot with the Japan rail pass and stayed in many funny, weird and nice places. Given the backpacking nature of those places I don't see them appropriate with young kids. Imagine staying with your kids in a capsule hotel or a love hotel with cheap overnight fees.
What form of cheap accommodation types would be ideal with children in Japan? I found some apartments online, but they require longer time frames to be booked, also limiting utilizing the full potential of a Japan rail pass. Regular hotels also tend to be rather expensive.