When writing in the capacity of an economist, one should never pitch an experience as “free”. Going anywhere has opportunity costs, especially if you have to pay several tolls on the New Jersey Turnpike. That said, if you want kids to see New York City as part of a road trip, consider Liberty State Park.
You can enter and park for free. There is a playground, picnic tables, and lots of trails. You can see the Statue of Liberty, Ellis Island and Manhattan across the river.
At the northern edge there is a memorial for those lost on 9/11.
If you have hours to spend, you could also pay admission for the Liberty Science Center. Currently, my family has free admission because of the ASTC Travel Passport Program.