I've met a few visiting MMM'ers - if anyone wants to meet forum folk, throw something in the NYC meetup thread!
Museum of Jewish Heritage is open until 8 on Wednesdays, with free admission.
There's also the Museum of American Finance, which closes at 4, but may be close enough for a lunch visit.
Museum of the American Indian is open until 8 on Thursdays, with free admission.
Fraunces Tavern is a museum/bar. One of the oldest NYC buildings, Revolution-era history.
Staten Island Ferry is fun, and free, as is the Brooklyn Bridge. In SI, you can walk a few minutes and get to a 9-11 memorial. In Brooklyn you can walk a few minutes to a park, a memorial, cool neighborhoods, another park...
Edit: There's also usually free comedy shows.
Time Out has a list that you can filter based on what activity you're looking for.