Generally you want to bring something that represents where you are from. Bring a small gift for each member of the family, and one gift for the family as a whole. Nothing extremely expensive is necessary.
They will also return the favor and if you give very expensive gifts, it puts the pinch on them to give something of equal or slightly higher value. Foods are nice but also give them something to remember you by after the food is long gone. Keep a gift in reserve also to give the family just before you leave. Some typical items are: picture books or calendars, ashtrays, coasters, chocolates, fragrant bath salts, t-shirts or jackets of local sports teams, stationary with famous animation characters (except Mickey Mouse or Snoopy etc., they've got tons of that), BBQ sauces; dijon or dark mustards, dressings like Ranch, mushroom, herbal, etc., pure maple or berry syrups, some special local jams or preserves (except strawberry or marmalade, they got tons), baseball caps, fragrant soap or shampoos, lotions, cosmetics, salad toppings/bacon bits, air fresheners, nuts, dates, dried fruits, flavored coffees and teas, key chains, postcards, flavored powder coffee creamers, recordings of TV or radio, marinades...etc, etc.
Be aware though that in Japan 4 and 9 are "unlucky" numbers, and especially older Japanese tend to be
superstitious, -- avoid giving any sets of 4 or 9.
I often bring some tea from a supermarket - it is cheap, light, and there are now several varieties of green tea that Japanese have never tried, such as with mandarin orange by Celestial Seasonings, or green tea with jasmine, blueberry, earl grey, apple, raspberry, lemon, ginseng or mint.
Somewhat less elegant but something quite practical I often bring for friends in Japan is a large bottle of aspirin, which is insanely expensive in Japan, and some Vicks vapor rub for colds.
It is very likely that your family will treat you very well if not lavishly, especially when you first arrive,
such as taking you to some expensive restaurants, so make sure to bring something more formal to wear like a better dress and dress shoes or high heels. If you're going during the hot season, be prepared for some rainy and very muggy weather - as well as some strong sun. Bring some sunblock with you too - it is also insanely expensive in Japan - and don't forget some deodorant also.
Try to help the family there, and be a member. Offer to help with the dishes, chores, etc. Perhaps you could help make some food from where you're from, like burritos or fajitas. Just take the powdered spices and some tortillas (they can supply the meat, veggies, etc). (The tortillas should be frozen or refrigerated as soon as possible after arriving).
Learn as much of the language and manners as you can. It will make things easier. See:
Plus learn something about where you are going - study some basic Japanese history, as well as the most interesting places to visit in the area. I did a homestay over 30 years ago, and was asked on a Sunday if there was any place I wanted to go - I was clueless. Some good sites to look at are:
Above all, relax and enjoy yourself.