The Nationals name comes from the fact that the team is located in the nation’s capital, in Washington D.C.
The Nationals name was actually officially the nickname of the American League Washington club for more than 50 years, though everyone pretty much called them the Senators. Later, the expansion Washington team who played in the American League from 1961-1971, was also the Senators. In 2005 when the Montreal franchise went south to D.C., they revived the Nationals name, “Nats” for short.