Hi Neil. Although I grew up in NYC, I'm not all that familiar with New Jersey. I do like mps69's suggestion, as you can easily take the PATH train into the City. I see that the Hampton Inn Dave recommended provides public transportation to Manhattan, according to the Hilton Web site. I wouldn't stay anywhere that requires you to take a car or taxi as those costs can nearly offset the savings of staying out of town.
I am more familiar with Westchester, and have stayed several times at the Hampton Inn that CruiseDad mentioned, as well as the Springhill Suites down the road, also in Tarrytown. There are also some hotels in White Plains that are near a MetroNorth commuter station, although they may be pricier.
December in NYC is tough as the City is packed for the pre-Christmas season. I've been wanting to go there myself, but weekends in my favorite Manhattan hotels are sold out.