Okay, I have some time to type now
Staying in NJ is only good if you don't really plan on going out at night... NJ Transit stops leaving Penn Station around 1:42 am. (Not that I ever take that train or anything
) Plus there is nothing to see or do in New Jersey itself (I hate to say) except go to the shore, but I mean the best beaches are way in the south of the state and there is too much else to see, I doubt you'd want to go all the way to Wildwood.
So that said, I would just try to find a cheaper hostel/hotel in NYC like Adam said. There is a lot to see in NYC especially if you are doing tennis also
If you want to save money, stay in one of the other boroughs instead of Manhattan (like Queens or Brooklyn) as those will be less expensive.
If you do want to explore DC, it is too far from NYC (I think) for just a day trip, and also too much to see. From where I live in central NJ it's about a 4 hour drive to DC, so from NYC it would be about 5 hour drive and trains probably aren't any faster. You'd maybe want 2-3 days there to do museums and monuments, but there's still no way to do everything.
Honestly it depends on how much tennis you plan on doing, whether a side trip to Washington DC would be worthwhile or not. Another option would be Boston, which from NYC is about the same distance
I think if you only have a week and you plan on doing several days of tennis, there is a lot to see within NYC itself