If you like theme parks there is a six flags in Springfield Mass. Also look at a place called 5 wits at Patriot PLace. They aren't in Boston but they are fairly close. If you don't like huge roller coasters try Canobie LAke PArk instead it is in nearby Salem, NH. I would not recommend Faneuil Hall market it is a tourist trap. The New England Aquarium is a nice place for a one day visit. Boston museum of science could take you a couple of days in my opinion. I don't know where you live but Boston museum of science has an annual pass and it gives you entry into many museums around the country. Look up the list to see if they are associated with a museum near you. Then you can get an annual pass and use it many times all year. Just walking around Boston is pretty cool.
I must interrupt here and say that, relatively speaking, Springfield is not "fairly close" to Boston.... at least not in my travel terms. It is about 90 minutes (not including traffic) from Boston to Springfield. Canobie Lake Park is a little closer at about 40 minutes (not including traffic). And I'm going to slightly disagree and say that Faneuil Hall is a tourist trap, but still fun... especially for lunch!
Five Wits at Patriot Place is a great suggest for a family with kids and if you are sports fans you can see inside the Patriots Stadium while there. But again, it is about a 30+ minute drive (without traffic) out there.
I echo the 5 Wits post. Very clever and fun. Patriot Place is great.
Take a tour of Harvard across the river...interesting history
USS Constitution is interesting
For young kids try the a Children's Museum. The aquarium is sorta meh compared to others. Harbor Islands are amazing. Ferry out to George's Island is a great way to see the city from the water
We might still have snow on the ground.
I guess I'm in a mood to disagree today, because I disagree that the aquarium is "meh".... have you seen the newly refurbished Giant Ocean Tank??! It is beautiful! If you take your time, there is SO MUCH to see at the NEAq! The Shark and Ray touch tank and the Edge of the Ocean touch tank are both great for kids (and adults!). The Museum of Science and the Children's Museum are both terrific.... but given the ages of your children, the MoS might be too much for them (although it does offer really cool Imax movies and has a planetarium).
Also in Boston there are history tours, walking tours, Segway tours (I realize your kids are too young for that though), etc.
I second the recommendation of the Duck Tours. If anyone in your group is into history, there is the Old North Church, the Paul Revere House, the Bunker Hill Monument, and just outside Boston in Salem is, of course the Salem Witch trials stuff.
And there are a few zoos within driving distance.... Franklin Park Zoo and the Stone Zoo..... and while I admit I've never been to the FP Zoo and had a less than stellar visit to the Stone Zoo (many years ago), it might be a good afternoon visit if your kids like zoos.
To get a good view of the city you can visit either the top of the Prudential Building or the John Hancock Tower.
If you have a car and don't mind driving, in addition to the things that others suggested, you could visit the National Seashore on Cape Cod or the mansions in Newport, RI.
If you do visit during the July 4th holiday, just keep in mind that EVERYTHING will be more crowded in the city and probably more expensive (hotel rooms, rental cars, etc). But it would certainly be an exciting time to visit!....................P