What gift should I buy?

Ideally something which is a famous or specialty product of your hometown or state. A photo book of your state would be nice, or some kind of unique food or seasoning.

Also, for Japanese people, the presentation is more important than the gift itself. Make sure it is carefully and beautifully boxed and wrapped.

I visit japan often and out of all the gifts, the most safest and widely appreciated has been handkerchiefs and biscuits. You can stock pile handkerchiefs to give out during your stay to other people also. Most people carry them around and have different quality for different occasion. Also the biscuits should be on the light side that is not too heavy in flavour. Also they love to accessorise things like mobile phones or key chains so maybe that is something you can buy for the daughter that is from your area.

So, to summarise, a couple of selection of biscuits so that the family can share and enjoy, as well as proudly sharing with neighbours over an afternoon tea.

Numerous handkerchiefs mens for papa, womens for mama and the Daughter as well as spares for later encounters.

A key chain or dangly thing for mobiles which are unique to your locality. So the Daughter can remember you by.

Above all, wrap it nicely, :)

Enjoy your stay!

Never try to pick any book, photo book written in English because they don't have any interest on those things except they are scholar or Otaku for American.

Handcrafts or small things are also not good ideas. Japanese products(including products made in China) are much better than things made in USA and they know it very well.

All I can recommend and actually with my own experience, food is definitely best. Not typical machine made macadamia Chocolate but local homemade Chocolate, Pie, candy sweets or something like tidbits taste good will be better.

Japanese in general love the Yankees. A Yankee ball cap or other paraphernalia can go a long way with the husband.

Japanese women love beautiful things: a delicate silk scarf or a fan is nice (anything handmade would be good)

And for teen, something very american, trendy and preferably designer!

and as mentioned, wrap and present them beautifully -- it reflects your thoughtfulness and care in selecting the gifts.

Maybe a book about the different states. They may be curious about what they are all like.