We stayed at the same place the Grid Iron Club did and loved it. I think it was the Millenium Hilton (across from Ground Zero). We stayed an extra night and I think the rooms were only $250/night around Christmas. It would be worth checking out to see if you could find a deal there. I was disappointed initially not to be staying in Times Square, but Lower Manhatten turned out to be the best option. A bit more quiet than Times Square. My 62 year old Dad said he felt safer walking around New York at night than he did walking around Waterloo at night.
We didn't have tickets for the Ground Zero Memorial, but could see it from our rooms. We did go into Trinity Church and it was really nice to see. We really liked Bryant Park, but that might be beause it was decorated for Christmas and less busy than Rockefellar Plaza. Go see a Broadway show. We saw Memphis. It was excellent. Ellis Island is pretty nice. Walk around a lot and just explore. Walking across the Brooklyn Bridge was more fun than I would have thought.
I went with my parents, brother, and sister since my other 2 sisters were in the band. I think most of us weren't too excited about NYC thinking that we wouldn't like it. But all of us really did (my Dad said that if he was 40 years younger, he would want to move there and my mom who hates traveling still can't stop talking about how great the trip was). Part of that was circumstance....New York at Christmas is pretty awesome and we were lucky enough to have perfect weather, but it was still a pretty great trip (with the exception of the football loss).
Bottom line, I would tell you not to skimp on your hotel. I am convinced we had a much better trip than had we stayed in New Jersey like we had originally planned. Other than the hotel (and the broadway tickets), New York really isn't that expensive. You can eat good lunches for cheap and most dinners aren't too bad. I had heard the subways were hard to get around on, but they were actually pretty easy. The people were really, really nice. My mom talked with a couple of police officers who had some pretty sobering stories about what they were doing on 9/11.