First let me preface one thing. There is always the "avoid tourist traps" warning. But some times you have no choice. Like The Eifel Tower is a tourist trap but if you go to Paris, how do you not go up the Eifel Tower?
1. After 9/11 Memorial, walk west to Broadway and then turn left and head north to City Hall. Stay on the right side of City Hall to get a nice view of the Brooklyn Bridge. You can walk halfway through the bridge and then turn back around. It is a nice walk with good views but if time is an issue, it can be skipped.
2. The east side of City Hall is Centre Street. Walk north on it till you get to Worth Street and then make a right. Make a left on Mott Street. On Bayard Street near between Mott and Elizabeth, there is the Chinatown Ice Cream Factory. Try some of their unique flavors. My favorite is Pandan.
3. Head back west towards Mulberry Street. After you pass Canal, you've entered Little Italy. Keep heading north on Mulberry till you get to Grand Street where you have Ferrara's Bakery. Tons of Italy dessert you take on the go.
4. Continue north till you reach Houston Street (it's pronouced House-Ton if you ask locals for direction) and then make a right and head east till you get to the corner of Houston and Ludlow Street. Go into Katz Deli and grab a pastrami sandwich. Moan like Meg Ryan did in "When Harry Met Sally". Also, you've just walked through most of the Five Points if you want to get historical.
5. From there, grab a cab. The surrounding neighborhoods there are more artsy and hipster. I'm going to take a wild guess and assume your mother is not interested in seeing men with nose rings and stuff. Excuse me if I'm wrong. Cab it to 23rd and 5th Avenue. There you have Madison Square Park and the Flatiron Building as well as the original Eataly (they opened a new one by the 9/11 Memorial which I haven't been to yet).
6. Walk north on 5th Avenue till 34th street where you have the Empire State Building, Macy's Heard Square (34th between 6th and 7th Avenue) and Keen's Steakhouse on 6th Avenue and 36th Street. In my opinion, best steakhouse in the city. Get the Porterhouse for 2. Really great whiskey selection as well. While I don't know their actual cut off in regards to dress code but I've gone there in jeans and sneakers on several occasions without any issues.
7. End the night at Times Square. It's not worth seeing during the day so best to come at night when all the lights are on full blast.
8. Carmine's in the UWS and Tony's on the UES are good. Tons of food. Just keep in mind that they're New York Italian food and not really true Italian food.
After Central Park and Museum of Natural History, what you're missing is Statue of Libery/Ellis Island tour which can be coupled with you "boat" tour even though that ferry doesn't really go around the city. Musuem of Art (I think she'll really enjoy it. They have an entire Egyptian Temple inside).
A tier below you also have the UN Headquarters and the Intrepid Sea Air and Space Museum. The Circle Line cruise is right next to the Intrepid so you can couple the two together. If you want to stick in that area, walk north along the Hudson River. They've complete rebuilt that pathway and it's very nice. Especially north of 59th street but beware, the department of Sanitation is at 59th street so it get a little smelly right around there but once you pass it, it's a nice walk. I also recommend The High Line. Best to enter at 12th street and Washington Street and walk north till you get to end at 30th street. This drops you off right at The Vessel. I'm not sure if they've opened The Vessel again after several people committed suicide from jumping off of it but it's still nice to see from the bottom too.
The other note worthy stuff to see in the city are out of the way and with a limited time probably not worth committing such a long travel just for one site. Cloisters (an extension of the Museum of Art devoted to Christian Renaissance art), Bronx Zoo (there is a Central Park Zoo but not as good as the Bronx Zoo), New York Botanical Gardens, Brooklyn Bridge Park to name a few.