The Red Hook Arts Festival and Celebrate Brooklyn both involve some awesome fun events, and PS1 always has interesting daytime things happening. I often organize my summers around making NYC more tolerable, but there are many interesting free or cheap oddities around that make you forget you're in NYC: The Cloisters way uptown, The Roosevelt Island tram, the various beaches (Brighton Beach and Coney Island are accessible by train, and this might be the last year to see the seedy, low-rent, Chris-L-chili-dog-pic version of Coney Island before it gets turned into Times Square. There are dozens of art-house cinemas (Film Forum and IFC usually have the best stuff, IMO), and while the Strand in the summer will be so full of cranky, smelly weirdos that you'll feel like you're in a live-action FOTchan reenactment, they have some of the most amazing deals on books I've ever seen. The NY Public Library usually has some cool literary stuff on display, and if you can use your student ID, Moma, The Met, and the Museum of Natural History are all worth checking out. Clubbed Thumb's Summerworks and Arielle Tepper's Summer Play Festival are probably the best deals on theater if you want to avoid schmaltzy Broadway garbage (I think they're like $15/ticket) and I'm happy to make specific recommendations for you.
All of that is Googleable.