Ok, I know you cannot stay in Boston and spend without going over $150. But there are inexpensive buses to Boston and DC - check out Bolt Bus I believe. With the bus and an overnight stay you could have a nice time for $250 if you stay at a hostel where prices go below $100 for the night.
If you want an adventure check out the Gould Library and Hall of Fame then try some West African food or find a great Dominican restaurant down by Yankee Stadium. The Gould Library and new library as well as Hall of Fame are in Bronx Community College. Here is a pic of Hall of Fame -
Hall of Fame for Great Americans, Bronx Community College, University Avenue at West 181st Street, University Heights, The Bronx
Less of an adventure but still fun is check out the new park at the south end of Roosevelt Island and walk along the water to see LIC and eat midtown Manhattan and how large the Queensboro Bridge is, then hop on the F train to Jackson Height for either Indian or Ecuadorian food. If you want to go back to the city take the F to 2nd Ave and check out rare films at the Anthology Film Archives. Walk up 2nd Ave and stop off at any of the many outdoor cafes and beers running all the way to Union Square/14th Street. If you still have time stop in at Beauty Bar on 14th for a drink in one of the salons chairs with large hair dryer piece over head.
I second many of the options given for Brooklyn. MoMA and the Met can be madhouses - but go early in the day (before noon) and it is much more pleasant. Investigate smaller museums - The Museum of the City of New York is gorgeous and has cool history about NYC. Near there are the Conservatory Gardens of Central Park and Harlem Meer. Less crowded than the southern areas of Central Park.