An option that would save you money is to stay across the Hudson in Jersey City, NJ. That's what I've done in the past. I've stayed at the Hyatt on the Hudson at Exchange Place. When I would price out comparable rooms in Manhattan, they were usually close to double the price. The PATH station is right across the plaza from the hotel door - two minute walk, maximum. That will take you into Lower Manhattan to the World Trace Center stop. From there, you can catch the bus, subway, or cabs to wherever you want to go in Manhattan...or just walk.
To see the Statue of Liberty, you may want to consider getting the ferry from Liberty State Park in Jersey City instead of Batterey Park in Manhattan. The lines to get on the ferry are much shorter from Liberty State Park. The trip just goes in reverse. Instead of Statue of Libery, Ellis Island, back; it's Ellis Island, Statue of Liberty, back. Although, due to Superstorm Sandy, Ellis Island is still closed for repairs - the Statue of Liberty just opened on July 4.
I'm not sure what type of stuff you'd like to do but some of the things I've done and would recommend are:
Federal Hall (
http://www.nps.gov/feha/index.htm) - George Washington took the oath of office here. It's run by the NPS and, if I recall, entrance is free (they do ask for donations). It's on Wall Street sort of kitty corner to the New York Stock Exchange.
St. Paul's Chapel (
http://www.trinitywallstreet.org/congregation/spc/) This is across from the WTC site. It's a 9/11 Memorial site and has museum exhibits dedicated to that. When you see how close it is to where the WTC building stood, it's kind of amazing that the only damage it suffered when the buildings collapsed was a tree that was uprooted.
Trinity Church (
http://www.trinitywallstreet.org/) this is at the top of Wall Street
Top of the Rock (
http://www.topoftherocknyc.com/) This is the observation deck (covers 3 floors) at 30 Rockerfeller Center. It's similar to going to the top of the Empire State Building but not as crowded...and you can see the Empire State Building from it. It's got great views of Manhattan and you can see as far out as the Statue of Liberty.
St. Patrick's Cathedral (
http://www.saintpatrickscathedral.org/) It's on Fifth Avenue a stones throw from Rockerfeller Center
Double Decker Bus Tours (
http://www.newyorksightseeing.com/c...ayLine_New_York_Buses&utm_campaign=GLNY-Buses) - These are a great way to see the city. I've done it a few times and when we took my nephews to NYC, they loved it. They're convenient in that they're hop on/hop off and stop at most of the big landmarks. The only exception is if you do the night tour (great way to see the city lit up at night) in which case it's just a straight tour with no on or off.
Battery Park (
http://en.wikipedia.org/wiki/Battery_Park) You'd catch the ferry to the Statue of Liberty from here. There are also several memorials in the park as well. The sphere from in front of the WTC is located in Battery Park with an eternal flame. There are great views of the Statue of Liberty from here as well. (The city of NY has an official website for the park but when I just tried to access it, my virus software had an alert that there's something wrong with it...so you might want to be careful if you go to it).
If you're in Jersey City, there's also a 9/11 Memorial along the waterfront. It's small but it's quite poignant. I believe they've also opened a bigger one at Liberty State Park as well in 2011.
This is all I can really think of right now off the top of my head.
on that day - cheaper, but only certain shows available. 
