Parking at hotels in D.C. can take a chunk of money out of your budget. And depending on where the hotel is in D.C., you'd likely need to use Metro anyway to get to the tourist sites near the Mall.
If you find affordable accommodations in Arlington that you like, I see no reason not to stay there--lots of visitors to D.C. do just that. You'll want to choose a hotel that's near one of the Metro stations. In Arlington, that would be Crystal City, Rosslyn, Courthouse, Clarendon or Ballston.
Look at the Metro website to see what the fares will be and to see if any of the fare passes suit your purposes. www.wmata.com Because you are a family of 4, it may turn out to be cheaper to drive into D.C. and park in a garage. A good resource for parking rates is www.bestparking.com
FWIW, during a power outage this summer, we spent a couple nights at the Best Western Iwo Jima located a few blocks from the Rosslyn Metro station. Not only did they have good rates, but parking was free and they have an indoor pool which your kids might enjoy in December.
Finally, you might want to post your query on the Washington, D.C. forum if you have not already done so.