... and if you stayed in Providence that would give you the option of taking the MBTA train right into Foxboro...
Now that might limit the amount of time you have to enjoy the whole "scene" before or after the game, as I don't know how early or late the trains run nowadays - and you might not want to limit yourself in that way - but as far as avoiding traffic, avoiding parking fees and just kicking back and not worrying about much, its the way to go
There's a good number of nice hotel options in Providence @$135 a night
And with a kiddo Providence can also be a help - if you stay at the Courtyard Marriott you're right across from the Providence Place Mall and plenty of restaurant options and a short cab or bus ride to the Children's Museum, Roger Williams Park Zoo, or other activities to kill some time before and after the game.