Japan's a big place. Where are you going? I'll assume Tokyo, since that's where most people atleast start. I'd recommend the Grand House Chang Tee in Ikebukuro, just because it's convenient, and on the Chuo line. Ofcourse, if you truly want to go the cheapass way and don't know anyone to stay with, you can just sleep in internet cafes (although, I doubt with children in tow).
The only warning I can think of is to pay close attention if your hostel has a curfew. After spending a few minutes longer than planned at the bar, I found myself spending the night in a Denny's after discovering that the hostel was already locked up.
I have no idea about kids.
As well, I have no idea about luggage limitations. Keep in mind that you're not paying much for a room (assuming you get a private room), you're in Japan (with typically smaller sized rooms), and you can always just find somewhere to wash your clothes here. I would strongly suggest bringing as little as possible. Especially if you plan to do any traveling within the country, and expect to lug your bags with you.
Whenever I've needed to get a hostel in Japan, I've used
http://www.itcj.jp/index.html. It's a central agency that interacts with a lot of smaller hostels, which is nice. Also, most of them will accept credit card payment through the website. Japan is very cash-oriented, so being able to put hostel payments on credit card can be convenient.
