Sorry for busting out that tired Electric Boogaloo joke again. It's a compulsion.
Last time, we had a look at the most obvious name for a DC team, and I proved conclusively that Senators is the only choice. In fact, I probably should have saved that for last, since there seems to be little point in bothering with the rest of the ideas. I shall continue nonetheless, but don't expect anything interesting.
Baseball Reference informs us that the Senators were officially the Nationals for fifty years. I don't know why, and I don't particularly care. No one used it anyway. The name is just as dry and traditional as the Senators, but without the resonance. Ask your average baseball fan who Walter Johnson played for, and he'll say "who?" Ask someone who's read The Glory of Their Times (which is as good as everyone says it is), and he'll say "the Senators." He won't say "the Nationals." The Senators v. 2 didn't bother with it, and neither should we.
Baseball in DC has a poll asking for your choice for the name of the 'Spos. At the moment, Senators is leading, followed by Grays and Monuments. I'll tackle the Grays another day, but where are people getting Monuments? I've never heard of a team with that name, and I hope I never do. I would approve of this name in one impossible circumstance: a colleague of mine had the idea to build a ballpark on the Mall, with the Washington Monument serving as a foul pole (Hi, George). The impractability of this idea detracts not at all from its brilliance. A team playing this park could be called nothing but the Monuments. Otherwise, forget it.
There are two traditional names that have not come up in any discussion I'm aware of, and I present them here only to show off my erudition (by which I mean my ability to surf around on Baseball Reference). There have been two teams known at one time or another as the Statesmen. As far as really Washingtony names go, it's better than the Monuments, but I can't think of a good reason to prefer it to Senators.
1873 saw a National Association team called the Blue Legs in Washington (no doubt over the objections of the owner of the Baltimore Canaries). People back then were incapable of coming up with totally rad names like the Voodoo or the Burn, so they mainly just named teams after what color their socks were. Hence the Red Sox, White Sox, Reds, Cardinals, and Browns. That worked out fine, but calling a new team the Blue Legs would be akin to me saying something like, "the New York Hilltoppers clouted prodigious circuit blows in the World's Series of Base Ball, to the delight of the bugs in the stands." That kind of thing can be cute for a while, but before you know it you're growing a handlebar moustache and listening to ragtime.
Next time I'll take on some Negro League names. This is going to be tough - so far I haven't been doing much but reciting what I dig up on Baseball Reference, which doesn't have Negro League teams. I hope my extensive research (a couple hours at a Barnes and Noble) has provided me with enough information on this topic.