You might be able to get an airport hotel for cheap - look around for hotel rates, and then lowball priceline (but beware some of the airport hotels they offer may not have a shuttle service). Take a nap, grab a shower, visit the museum than meet your brother and head off to wherever you're going. The room also gives you a place to dump your stuff for a few hours.

Where are you going once you meet up? If you're staying in DC, you might just want to head downtown and meet up there once he's in town. If it's an age/get around thing for either of you, the hotel option is a good idea. Otherwise you could buy a one time entry for a lounge pass ($50 - I'd rather do the hotel room in your situation, personally). You can't leave your stuff and walk around, the one time pass is really one time and the lounges are really not a good place to hang out for long hours. You'd do better to find a quiet gate where you can lay across a few chairs, IMO.