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I'm not too surprised about those 4 stations that you mention that have long-term parking. Franconia-Springfield and Greenbelt are at opposite ends of Blue Line. Huntington is the VA end of the Yellow Line, and the Reston station is the VA end of the Silver Line.
I'd definitely compare rates between the Metro long-term and the Reagan Airport long-term parking, and check for security too.
Donna
Long term parking at DCA is $17 a day. It looks to be considerably cheaper at the 4 Metro stations ($4.95 to $8.95) but as said, spots are limited. If you are just going in for the day, you can park at any station and pay a daily rate from $3.00 to $8.95, Saturday and Sunday is free. The parking lots do not accept cash, you need a credit card or a SmarTrip card.
I'd stay at a hotel that has free parking, then just drive to the nearest Metro station to go into the city.
You might want to consider the Baltimore-Washington International airport both for long-term parking and taking the train to NYC and for day trips into D.C. on the commuter train. Long term parking is more like $8/24 hrs. Many hotel shuttles will drop you off at the BWI Amtrak Station for the MARC commuter train (and you might be able to long-term park there as well). It doesn't take much longer from BWI Amtrak to DC Union Station than from the bottom of the line in Springfield/Franconia. And you are an hour closer to NYC.
If you are going down to Mount Rushmore, you should make time for the Crazy Horse Memorial. It is an amazing piece of art and the history depicts an amazing amount of information for the area. If you like hiking, Bear Butte is beautiful too.