I second the 930 Club idea, provided there's something good playing that Mom won't mind listening to. While I wouldn't call the area "unsafe", I wouldn't recommend 14 year olds treking there on the metro themselves. The site is http://930.com.

Another area outside Georgetown or U Street could be Old Town Alexandria -- depending on the weather it can be really beautiful and a lot of fun. There's also a cool store on King Street about 1 mile south of the water (I can't remember the name) that lets you pick out ceramic plates/mugs/jewlery boxes/etc and paint them. Might not work out with your timing (you typically have to wait 1 week to have your artwork sent through the kiln), but they might ship so it could be another rainy-day activity.

Or, you can venture to Bethesda, which is also chock-full of small shops and a lot of great eateries. I know I've passed a couple of vintage/consignment stores in the areas. Friendship Heights has some good discount-store shopping, too (Lohmans, TJMaxx, Filene's Basement).

As for places to eat, Sushi-Ko up on Wisconsin (within walking distance from Georgetown) is a hip place. Two Amy's which is a little further north off of Wisconsin (near the Cathedral) also has a really comfortable and cool feel, and has the best pizza in DC. A little pricier can be Zatinya, which is downtown and serves small tapas dishes which can be fun in a group.

A GREAT source for more food information is Tom Sietsema's chats which you can find (with some searching) on Washingtonpost.com. Tom Sietsema is the Washington Post food critic and I know has answered the "where to take teenagers" question hundreds of times, so it might be worthwhile to sift through his archives.

Good luck! Feel free to PM me if you need any additional information