If you arrive a couple hours before the game, there will be plenty of spaces (and tailgaters) in the official lots. The Cell is right off the Interstate and surrounded by official lots. To see the neighborhood itself, you'd have to go another 2 blocks north or west. Bridgeport is largely residential (mostly middle- and working-class families, though there's been a recent influx of homes that cost $1 mil. and up) with some good restaurants and divey bars. Grandstand (a huge sports apparel etc. store) is on 35th and Wallace.

In addition to the "Kid's Day" stuff you mentioned, the Sox have the Fundamentals Deck over the LF concourse. Definitely check that out. If he's going to run the bases, keep in mind that you have to line up before the game ends. Also, check out the wide variety of foodstuff available at different booths around the concourse. If you're interested, I think you can get a "first Sox game" certificate from Guest Services inside the park.

As far as tickets go, the prices and locations are easy to understand compared to most MLB parks. Where ever you choose to sit, your son's ticket will be $1. For adults on Kid's Day, the cheapest tickets on the lower level are the bleachers ($31 each). For an extra buck ($32), you get a regular OF seat (not a metal bench) in lower reserved. The cheapest tickets overall are upper reserved ($17). Upper box tickets (roughly between the bases) are $21. You can access Fundamentals from either level but can't access the lower concourse with upper deck tickets.