Maybe skating? There will be few skate links build during winter in Tokyo. You can rent shoes but have to bring gloves. Last year, Tokyo Sky Tree and Akasaka Sacas (shopping mall in Akasaka) built skate links.
If you are visiting in early January, Tokyo Disney Land and Tokyo Disney Sea will have new year decorations. The decorations are inspired by traditional Japanese Oshougatsu (New Year) styles, so it might be fun. Micky Mouse and Minnie Mouse wear kimono.
Not particular to children, but this article may help you:
- 10 Best Things to Do Limited in Winter in Tokyo
This one is not particular to winter but also may help you plan the trip:
11 Best Things To Do with Your Children in Tokyo