This is a terrific old map from 1867 that we dug up at the Library of Congress. The land on top of which St. Elizabeths Hospital was built had been referred to as St. Elisabeth since the Colonial times and the hospital had informally taken the area’s name ever since.
Source: Library of Congress
Take a look at the approximate area today on the Google Map. Some of the roads have the same names and you can see some lots are the same.