I've found that the best place to park for long jaunts out of DCA (especially if you live in the western suburbs of NoVa) is the Ballston parking garage in Arlington. Take the elevator into the Ballston Common Mall and use the enclosed walkway from the mall into the Metro Orange Line station. Then transfer to the Blue Line at Rosslyn for DCA.

Ballston is the cheapest overnight garage parking in Arlington that I've found. Fees are $8 max per day Monday-Friday and $1 Saturday and Sunday (cash only) and overnight parking is permitted since the garage is operated by Arlington County rather than by Metro.

There's a map and more info here.

Alternatively, there may be some free street parking available in parts of Arlington accessible by Metrobus or rail (side streets near the Fort Georgetown Apartments in Rosslyn come to mind) but be careful that it's not permit-only or time restricted.