A very helpful and important list!
  1. Walk the Brooklyn Bridge (but make sure you've planned which restaurant you're going to after).
  2. High Line
    Suggested by @ChrisK
  3. Chelsea Market
    Suggested by @ChrisK
  4. Brooklyn botanical gardens
    Suggested by @jessicasfera
  5. Broadway musical
  6. Wonder Wheel in Coney Island
    Suggested by @second_lunch
  7. Circle Line and see things.
    Suggested by @donnie
  8. Tea at Bergdorf's if that's your mom's kinda thing
    Suggested by @carey
  9. Somewhere iconic and fancy like the Russian Tea Room
  10. Tenement Museum Tour (not on a hot day, though)
    It's expensive, but she will pay so you both can get your learn on.
    Suggested by @TheAnne
  11. Free yoga in the park (Battery park Sunday mornings, Prospect Park/Bryant Park Thursday evenings, etc)
    Suggested by @Kat
  12. Free Tours by Foot NYC : if you're both into learning the history of certain neighborhoods (they do great food tours) and affordable.
    Suggested by @Kat
  13. The Bronx Zoo
    Suggested by @dudleyjoshua
  14. Watch HGTV shows in your apartment.
    Suggested by @carlye