The best kid-friendly antidote for hot, humid Japanese summers is "kaki-goori," or what they call "shave ice" in Hawaii.

My personal favorite flavor is "mizore," or diluted honey syrup, but you can get fruit flavors as well. Kaki-goori stands can be recognized by their blue flags with a big red starfish-looking kanji (the kanji for "ice") flying outside.

I second the recommendation for Hakone, simply because of the variety of transportation options on the Free Pass.

A genuine samurai castle, such as the one in Himeji, might go over well, too.