abmj-jr suggested Hakone as somewhere to visit and I think that might be an ideal excursion for your family. There is a ticket called the “Hakone Free Pass” which allows you unlimited travel on the different transport systems of Hakone, which most people travel as a circuit (takes roughly 4-4.5 hours)
The trip from Shijuku to Odawara should take between 1.5-2 hours. From there you get
The Hakone-Tozan Line mountain railway to Gora and begin the circuit. At Gora change to the Hakone-Tozan Cablecar to climb the steep incline up to Sounzan. Then the trip becomes more fun when you board the Hakone ropeways up past the boiling sulphur pits of Owakudani and then on to the edge of Lake Ashinoko ***. There are about three plastic clad ships made to look like galleons and pirate ships (they wouldn’t look out of place in Disneyland) and these cross the lake (you can get a splendid view of Mount Fuji from the deck on clear days) and once across there is some lovely mountainside scenery to explore before getting a bus back to Odowara or Hakone-Yumoto (I swear my partner and I once saw a hummingbird on the shore of Ashinoko, but they aren’t native to Japan).
The main reason that I would be keen to take children here is that you get the chance to do a lot of walking amongst some fantastic scenery, yet the transport options mean that there’s no need to carry a little one very far once they succumb to fatigue.

Also, there is a Universal Theme Park near Osaka/Kyoto http://www.usj.co.jp/e_top.html

*** I remember that there was a craft shop here that let you paint onto ceramic/pottery pieces. We didn’t do this ourselves, but we saw a Japanese family sitting outside in the sunshine getting very involved in their projects.

P.S. I've always avoided Japan in the summer, but there is one reason I'd visit purposefully at this time, and that's to see the fireflies in early July. The only places that I know of where you can see them (but I'm sure there are many more) are near Mount Takao (one hour west of Tokyo -makes a lovely day trip) and around Kamakura - apparantly there is a park on the top of a mountain there where they can be found.