I do not believe that leaving bags in lockers is an option at Logan, and the Virgin check-in counter does not open until mid-afternoon.

Boston is a fantastic place to spend a day with kids: there's a lot to do, the city is great in summer, and most of the things you want are within easy walking distance.

I would focus on the Public Garden/Boston Common area (the Public Garden and Common are adjacent to each other). The Public Garden is a terrific city park with lots of trees, paths and benches, squirrels who are confident enough to walk right up to you looking for handouts, a Swan Boat ride that is not to be missed, and generally an ambience that is the envy of most other urban parks on the planet. The Common is not as pleasant (more of a big open space than a treed and flowered park), but still a nice place to be. Next to the Common is Beacon Hill, a wonderful place to stroll around (especially the side streets) and to see 18th and 19th C. American architecture.

It's also an easy stroll from this area to the Charles River Esplanade (another place to relax and enjoy the city), or to the city's best shopping along Newbury and Boylston streets.

For logistics, the least expensive and most convenient thing is to park at the Boston Common underground garage. However, finding the garage (and later, finding the way from the garage to the airport) might not be terribly easy for those unfamiliar with Boston. An easier, though more expensive, option would be to park the rental at Logan's central parking lot, then come into Boston by taxi, subway, or water shuttle. (One might also check with the car rental place -- maybe they can store the luggage for the day).