You're probably going in on 80, so I'd stay at the TA that's just past the PA/NJ line in Jersey. Yeah, you'd have to get up early to go in, but my first drop Monday is 0400, so why should you get to sleep?
If you want to be as close as possible I would suggest the Vince Lumbardi rest area at the top of the Jersey turnpike. They have a Drivers Lounge in the back of the Building. Not the best place or the worst,but closest. The other options mentioned are good too,or you could stay at the Petro and go to the Vince sunday night.
rt 440 has quote a few large parking lots to park in where no one will bother you. The stop & shop/ Walmart center has large areas to park by the loading docks. It's basically one long wide road. There is a movie thearter, shoping & quite a few places to eat in that location.