That’s kind of a funny part of DC… not a lot of restaurants of note south of the mall except a couple near the DC Waterfront.
Your best bet will be to hop on the metro. Two good areas to look for restaurants are Penns Quarter (near the Navy Memorial stop) and Dupont Circle.
In Dupont Circle, you can’t walk more than 3 store fronts without walking past a food or drink establishment of some type and if you fire a cannon in this area you are most likely to hit a Starbucks. Really good restaurants of all kinds: for seafood try Pesce on P St west of the Circle, for beers try the Brickskellar, there are several good Italian places. For some of the best sushi in town, walk a couple blocks east of the circle to Sushi Taro on 17th St. Also walkable from the Dupont Circle stop (north exit) is Adams Morgan. Kind of a hip-happening part of town if you’re into that. Plus several good Ethiopian restaurants, if you’re an adventuresome eater – they are worth the trip. My favorite dish is the spicy red lentils.
Penns Quarter also has a mix of restaurants and is near the Verizon center so there are several sports bar type places, but the mid to upper end of the scale in this area can be quite good, plus the upper end of Penns Quarter is the DC Chinatown. Recommend: for a very nice meal with live jazz piano background and a great bar for what in DC is a very reasonable price try 701 (at 701 Pennsylvania Ave. – right next to the Navy Memorial Metro stop), for pizza try Matchbox (my favorite pizza in DC), for really goood tapas and a fun atmosphere try Jaleo.
When you’re over near the White House – go to the street on the east side of the Treasury building and look for the Old Ebbitt Grill. This place has a true DC politico atmosphere. They also have a decent Sunday brunch, a great bar and if you’re into it usually a very good selection of oysters.
There are a ton of other good restaurants in town. too many to put in one message, but that ought to get you started. The restaurants at the Smithsonian are okay and better than they used to be, but nothing to write home about. Also, you will have to go through security as you go into each museum, depending on how busy they are this can take a bit of time. You might want to plan to go to one museum for the morning, then out for lunch and to another museum in the afternoon. I used to be able to give a running tour of portions of about 5 museums in one day, but sad to say that era is over.
Hope that helps… there’s a lot of good eating in DC, enjoy!