Does anyone know of a cheap hotel in and around Boston, U.S for a family?

If I were you I'd be sure to check out TripAdvisor.com. It's a great resource for reviews about hotels in and around the Boston area. Cheaper rates will always be found outside of the city in places like Brookline (that's where I live) and Cambridge (across the river from Boston Proper) I recommend staying in Brookline if you don't mind a slightly longer commute (like 5 short stops on the green line) Don't stay past Coolidge Corner though or things get harry. Look into the Courtyard Marriott (across the street from Coolidge Corner T-stop), Holiday Inn Brookline (across the street from St. Mary's T-stop) and Beacon Inn (1 stop outside of Boston Proper). Each are very different but the rates are lower.

Also think about staying at some of the awesome Inns in the Back Bay which are budget priced. Grab them early though--they tend to get booked up quick.

http://www.463beacon.com/
http://www.charlesmarkhotel.com/