Don't go cheap inside DC - not all neighborhoods are safe. Holiday Inn Express or Hampton Inn are good beats for safe, cheap and clean. Try Arlington or Fairfax in VA...Try Landover, Chevy Chase or Greenbelt in MD. Or you could try Columbia, MD which is about 1/2 way between Washington and Baltimore.

Balitimore is another option if you want to visit some sites there. I think we pricelined for $45 in the Inner Harbor. The commuter rail from Camden Station will drop you off at Union Station (maybe 4 blocks north of the capitol building) for $8/person.

Harrington Hotel.It's downtown and you can walk anywhere you want to go or take the metro.We had a great time there.

Check out www.airbnb.com

I dont know "around" DC but i wouldnt suggest Red Roof Inn in Chinatown, although it is cheap. Noisy, humid, old building, not as clean as expected.

Marriott Crystal City

You can also stay outside of the city (Alexandria, Fairfax, etc) and take the metro in to the city. Hotels tend to be cheaper out of the city. So if you were comfortable doing that with a 1 year, could save money and put you right in the heart of the city.

Try the International Youth Hostel in Washington DC. It is a cheaper option than most hotels. Not all rooms are dormitories. Some privates are also available in most hostels.

Google Hostel Reviews and look up Washington. Read them and decide if you want to stay there and see if they have privet rooms available.