I would say if cost is an issue, you should consider buying a bicycle and living in the Capitol Hill or Eastern Market areas. Lots of young hill staffers in the area, rent (compared to the rest of the city) is reasonable. Cost is funny here though--though neighborhoods have different average rents, the real driver is proximity to a metro stop. So you can live in a great area, but if you're a 20 minute walk to the closest metro, that will do a lot to drive down the rent. 1 Bedrooms are going to run in the 900-1800 range, depending on where you are, but most will be in the 1300-1500 range.
As I'm sure you all know, but just to say it again--Georgetown law school is NOT close to Georgetown the neighborhood, which is where the undergrads are. Georgetown the neighborhood is a bit shi shi, very very very expensive, and NOT near a metro stop (I know that seems to go against what I wrote above).
Most popular neighborhoods: adams morgan ($$), Cleaveland Park/Woodley Park ($$), Columbia heights/Mount Pleasant/Petworth ($), U-Street ($$$), Georgetown ($$$), Eastern Market/Capitol Hill ($).