Hourly cars parked strategically around Boston such as FlexCar and the like generally run $8-9/hour, but are gassed, parked and so on and there's no muss or fuss about going to a rental counter in the middle of nowhere and returning to that location when you are done. Even if your Portsmouth trip is a total of 10 hours, getting a family of 4 up and back for $80 isn't bad and is competetive with standard rental rates when you consider gas & parking.

As to your departure from Logan, while taxis are expensive on the way in because the tolls & fees are all front-loaded, on the way out they are much cheaper. Figure $25+tip.

Downtown Boston parking during the holiday week is both expensive and close to non-existent. Either you will be parked way out of the way with very long walks or you will pay a fortune ($35-40/day) for secure, well-lit and convenient spots. And, that's if they can be found at all.

I would strongly advise you not to drive a car in downtown Boston because it's going to be costly & frustrating and you don't need to travel somewhere for that.