There are 5 of us arriving on the flight from Edinburgh on the 1st December at about 7pm and are wondering if there is any transport from the airport to jersey city , New Jersey and how much would it cost

I agree on the rental car, however, the following info is from an online source: NJ Transit operates a train from Stewart International Airport to Jersey City every 2 hours. Tickets cost $10 - $15 and the journey takes 1 h 8 min. Alternatively, you can take a bus from Stewart International Airport to Jersey City via Stewart International Airport, New York, Ny, and Port Authority Bus Terminal in around 3 h 20 min.

I would rent a car & drop it off in New Jersey. Otherwise you can take a bus to New York City & then a train to New Jersey. A car (SUV) rental will be much easier, faster, and not much more expensive given that you have 5 people.