We drive a lot farther....but we always come in the day before. I use Priceline to get a cheapie hotel room (usually under $35) for the night, and have no worries about missing the boat. 2 port area hotels offer a free parking/free shuttle deal with an overnight stay--the Radisson and Quality Suites (you leave your car in the Radisson lot either way). Parking at the port is $10 per day. However, the port authority lot is fenced and monitored, while the Radisson is an open (less secure) lot.
Technically, Priceline rooms are for 2 people....however, most hotels are "children free" and have no problem with kids in the room. I always call ahead and make sure we have a room with 2 beds (Priceline can give you a king room).
I know you're close....but car trouble, bad traffic, etc. can still happen.