You can probably fit it in, but it will be tight.
You didn't (originally) mention if this is an international connection (and in which direction), nor whether it's a weekend or weekday trip (important for traffic and the subways), so I'll assume the worst: flying in from an international location to NYC and connecting to a domestic flight at 6:28am on a weekday.
An eight-hour layover, minus two hours for security before your connecting flight, minus two hours for passport control and customs, leaves you four hours. Getting anywhere in Manhattan by public transit (which, by the way, operates 24/7) is going to be around an hour and a half each direction. That leaves you with an hour to enjoy Manhattan, which is nowhere near enough time to get to the statue and back.
However, if you don't mind spending some extra money, a decent way to see the statue is from the Staten Island Ferry. Take a taxi/Uber/Lyft from the airport to Whitehall Terminal in Manhattan (about an hour on a weekday, less on the weekend), then board the ferry (half hour wait + half hour ride, in the worst case), then get off the ferry and take a taxi/Uber/Lyft right back to the airport (another hour on a weekday). If traffic isn't too horrible, that puts you back at the airport at 11:30, with plenty of time to spare.
Since this is a domestic flight, however, you have more options. If you take the AirTrain (red line) from your terminal to Jamaica station, then get on a Manhattan-bound J train all the way to the end of the line, that should put you within walking distance of the ferry station. Then you can just take the ferry to Staten Island and back (they make you get off in Staten Island but you can turn around and get right back on the same ferry). The total cost would be $16 per person for the round trip (less if you get a MetroCard but you probably don't want to deal with that at 6am after a flight to save a few cents per person) and the time for the trip is around three hours on the subway, one hour on the ferry, plus some time walking and waiting at the terminal.
You asked about going to Central Park. That is at least half an hour by train from the ferry terminal. I'd choose one or the other rather than try to do both.
A quick note: Since you'll be travelling around Manhattan in the middle of the morning rush, you should know that people who live and work in Manhattan are always in a hurry. If you need to ask for directions, the proper way to do it is to say to someone random "is the ferry terminal that way?" Saying "good morning" or "excuse me" is considered wasting valuable seconds of their day. (Can you tell I've had to deal with Manhattan before?)