The menu isn't New England, but Orinoco, in the South End and more recently in Brookline Village, has delicious, casual Venezuelan food at very reasonable prices. The wait at the South End restaurant is long--no reservations--but I haven't been to the Brookline location yet, it looks bigger so perhaps not a problem. they do take-out. web site:
http://orinocokitchen.com/home.cfmChinatown is always kid friendly. Dim sum is always a good choice on a budget. I like Chow Chau City, they have dim sum weekedays at lunch upstairs, and all day on weekends.
If you go to the Children's Museum, there's a place around the corner for lunch that I am fond of, the Channel Cafe at 300 Summer Street. So-called because it's just off the Fort Point Channel, an inland waterway in the city. Nothing fancy but interesting and tasty salads, soups, sandwiches. Added bonus, part of the space is an art gallery featuring the work of artists who live in the building. Another nearby place I like is Lucky's, on the corner of Congress and A Streets. Good comfort food, though I'm not sure I'd consider it kid friendly, maybe for lunch. at night there's live music and an active bar scene.
For picnics, now that summer is here, the farmers markets sprinkled around town can be a good bet. In central Boston, I frequent the Government Center market on Monday and Wednesday, and the Copley Square market on Tuesday and Friday. They are sprinkled around Boston and Cambridge, so ask wherever you're staying if you have one nearby. In addition to produce, there's at least one bakery often selling sandwiches, savory croissants, etc. in addition to bread, pastries, or pies, and there's someone selling fresh goat cheese at Copley. the season is still early here in New England, esp. for fruit, but there are excellent strawberries right now.