Good idea. I have used a fare like that in on Metra in Chicago.
They should also consider a non-family group rate, either just on weekends or also off-peak weekday. They do this in Salt Lake City: any x people (4, I think) can travel together for a cheap rate. Like a family fare, there is just one actual ticket, so the group has to stay together, which I assume keeps many daily commuters from using it to save money.
Both are good ideas to get more people on CR on trains that are currently far from full, because they make the price more attractive to people who might otherwise drive and park together in one car. Should invite Salem, Lowell, Rockport, other places that have both stations and tourist attractions, to advertise in SSta, NSta, and BBY, and the museums that are near those stations to advertise at outlying stations. Basically invite them to make posters that the T will put up free: take the family to the Children's Museum for just (admission price) and $10 weekend train pass, no parking...