Where can a family stay in Tokyo?

S MORRIS, BALWYN NORTH

The best and simplest solution to your accommodation is Airbnb airbnb.com.au. I used Airbnb in Tokyo last August and it was a great solution, and there are plenty of two-bedroom apartments that will cost you heaps less than two hotel rooms, with more space and the ability to prepare some of your own meals. 
In  Tokyo you might look in the Ebisu district. This is one of the areas where I stayed and it's a great neighbourhood with lots of places to eat right on your doorstep and convenience stores that sell milk and other Western-style necessities that are not a major part of the Japanese diet. Above Ebisu Station is a shopping mall where you can pick up pre-prepared meals and salads and they're fantastic and great value. Ebisu Station is on the Yamanote Line, the circular metro line that loops central Tokyo and it's handy for sightseeing. Be brave, the Tokyo metro system is not for faint hearts but there are always station attendants who will help you out. 

Ebisu is fairly bustling and if you want somewhere quieter, just to the north is Daikanyama​, a high-class residential district and a really lovely area with proper sidewalks and funky shops and cool cafes. The only negative is Daikanyama Station, which is not on the Yamanote Line but it's not exactly a punishing walk to get to Ebisu Station. Both these areas are convenient for the N'EX express train service from Narita Airport, which will get you to Shibuya Station from where it's just a short taxi ride.

In Osaka, the Tennoji or Namba districts have been recommended as good areas to look for accommodation. 
New Year's Day is a big event in Japan, when many visit Shinto shrines to offer prayers for the year ahead but apart from that, there's really not too much to choose between your departure dates. 

Book ASAP, prices will not go down, unlike our dollar.