If you have not reserved a ticket to the statue of liberty by now, your best option is the ferry. You can take an elevator to the base, the steps up to the crown might be hard for a toddler.
If you go to Times Square, just be aware of the pan handlers dressed up as cartoon characters, they take a picture with the kids for a $1. Don't hold out a 5 & expect change, it will be gone in a second.
I just walked across the Brooklyn Bridge for the 1st time, living here for 15 years. It is a very cool experience. When you get to the other side there is a park and the docks. You can either take a ferry back over or the subway. Try to walk over early in the morning, we did that but walked back after having lunch & it was a nightmare of people. It is some of the best of views of the city I have seen.
There is also the Intrepid, if I remember right this has alot of interaction for kids to take part in.
Central Park is a good stop for kids. There is a small zoo there, I haven't been there but might be worth it. There is also row boats, carousal. I think it would be too early for the ice skating rink.
The Bronx Zoo is incredible. It is free one day a week, I think Wed, be prepared for long lines.
There is also the Brooklyn or Bronx Botanical Garden. They usually have some sort of exhibits going on.
When you say 9/11 memorial, there is alot there. Some people don't realize there is a Museum underground, it is VERY WELL done. I don't think the 2 year will get it but kids only take in what they can handle. I was worried about my nieces & nephew, they were 6, 7 & 8. The 6 year old boy liked to see the explosion over and over again. They put the really bad parts in little cubes so you don't have to see them if you don't want to. My brother didn't realize, had one of the kids on his shoulders and went in & did a quick U turn. When they did the museum,they wanted you to see the strength and the love that made the city come together. I think they succeeded. The memorials themselves, it isn't really somber place, peaceful. Kids do run around all the time. The freedom tower is open, a long line even with tickets but worth it. Pay the 15 & get the ipad, it will sync up to where you are and what you are looking at, you can then zoom into neighborhoods & bldgs and learn more about them. The subway stop has a new mall, neat bldg to see and walk around the mall. Some people may be bothered if your child starts screaming, some won't....develop the NY attitude and don't worry about it. You can't please everyone, do what is right for you.
You may not be able to take the stroller into some of the places. Just make sure it is a light one if you do use it. Many of the subways don't have elevators. If one of you carry the baby you might be better off. My brother usually put his kid on his shoulders.
I think little italy might be good with a toddler, great restaurants, gelato & bakeries. Don't worry about taking a toddler into most of the restaurants in the tourist areas, they are very used to them. As long as you eat, pay and leave a tip, they are happy. Do look at your bill, alot of them add in the tip in the tourist areas.
Going from the upper west side to downtown and back in time for a nap with all the lines may be harder than you think.
Let me know if you have any other questions
Mary