Boston is a great city for kids. During N4, if you have:
Top of the Prudential Center to the Skywalk (accessible from inside the Prudential mall). Cost $2 kids, $3 adults.
Christian Science Center's Mapparium. Located in the The Mary Baker Eddy Library, view an inside-out version of the world in 1938, done in stained glass; cheap and nearby.
Ride the T. Historic displays are at Boylston & Park Streets. Adults $1.25 (exact change in coins if you use the Prudential stop in the mall), kids 5-12 half-price.
Half a day
Go to the park. There are 3 parks only a short distance away. You can go 2 blocks north to Commonwealth Avenue, go about 6-8 blocks east on Boylston to Arlington (or take the T) to Park St and visit the Boston Common, or from the Boston Common go 4 blocks north on Charles Street to the Charles River Esplanade.
Swan Boats. Take the T to Arlington St, and walk about a block to the Public Gardens. After the boat ride (about $2 for adults, $1 for kids) take a walk over the world's smallest suspension bride, and walk around the pond, past the duck island (as seen in “Make Way for Ducklings”), and visit the statues of Mrs. Mallard, Jack, Kack, Lack, Mack, Nack, Oack, Pack & Quack.
Visit a playground. There's one at the Common near the Frog Pond (Arlington St or Park St stops) (bring a change of clothes--the kids will wade), and one by the ocean (Columbus Park)and near bocce courts (Haymarket or Aquarium, plus a bit of a walk), amongst others. If you're flying in/out, visit Kidsport at Logan.
A full day (and a bit more money to spend)
Walk the Freedom Trail. Start at The Boston Common, and follow the red line in the pavement. If you're not up to walking, choose a few spots: Paul Revere's House & the
Old North Church (North End), the USS Constitution, aka Old Ironsides (Charlestown Naval Yard), or the State House.
Visit a museum. (We have plenty; this is only a small sampling.)
Children's Museum, 300 Congress Street, Boston, MA, across the Fort Point channel from the South Station T stop (look for the giant milk bottle).
New England Aquarium, Central Wharf, Boston, MA. It has it own T-stop (Aquarium) on the Blue Line.
Museum of Science, Science Park, Boston, MA. It has it own T stop (Science Park) on the Green Line.
Franklin Park Zoo, 1 Franklin Park Road, Boston, MA.
The Arnold Arboretum of Harvard University, 125 Arborway, Jamaica Plain, MA
Museum of Fine Arts, 465 Huntington Ave, Boston, MA.
Institute for Contemporary Art (across the street from the Hynes), 955 Boylston St., Boston, MA.
Isabella Stewart Gardner Museum, 280 The Fenway, Boston, MA. If your name is Isabella, then your admission is FREE.
The Sports Museum, Fleet Center, Boston, MA. Take the Green or Orange line to the North Station
Bug Bugs exhibit, at the Garden in the Woods in Framingham. Sculptures of 12 different insects, in humungous sizes, are scattered along the woodsy Big Bugs trail. You'll need a car to get to this one.
Faneuil Hall Marketplace (aka Quincy Market). It’s a historic site, and a
place to shop and eat.
Filene's Basement, 426 Washington St., Boston, Take the Orange or Red line
to Downtown Crossing. The first & original offprice store, bargain galore!
Barnes & Noble, 395 Washington St, Boston, MA. Books.
* Many thanks to Lisa Hertel for this list of things to do in Boston with your family