The Woodland stop of the green line is fairly close to the Mass Pike (route 90) and over night parking is permitted there.
The Boston Common Garage is walking distance to Arlington stop and is one of the more reasonable priced garages in town. They will certainly be cheaper than what your hotel charges. Weekend rates are cheaper than weekday rates. I don't know i f they will charge a weekend rate on Thanksgiving day.
Woodland or Riverside Station.
I don't know where Wurtsboro NY is, or where in NY you're starting, but unless you're driving somewhere else after Boston it might make most sense to look into your question on your end not the Boston end.
If you're going to take the train in you may as well take full advantage of that and start no where near Boston. And by the same token if you're going to drive close to Boston you may as well get the most out of that drive all the way in.
Back Bay station, where most trains from way out will pass though, is close to Arlington station by the way. Most likely 'close enough' so you wouldn't have to worry about getting to Arlington via train from there.
although now that I think of it, depending how many people we are talking about, taking the train in from far out might not be good financially. You'd have to check on that.
If you are taking the train in, park in official lots and not on random streets or retail lots near T stations The retail lots do/may tow and the residents on streets clogged with T riders may nor may not be nice to your car.
Boston Common garage is probably at least $20-$30 a day (not sure if they have extended stay rates), whereas Woodland Station is $6 a day, a few minutes off the Mass Pike, and the T into Arlington is about $2.50 per person. I'd recommend getting a 7-Day MBTA pass if you think you'll be exploring Boston via the T. You can use it as many times as you like within 7 days.
bestparking.com is a great site.