While Tecumseh was an enemy of the US while alive, his name was well known. The first ship named after him was a Canonicus-class monitor during the Civil War, some 50 years after Tecumseh's death. The other Canonicus monitors were named after other Native Americans or places with names derived from Native American words (generally - the first was renamed Ajax from Manayunk after the war). So, 50 years after his death the name was still known.
Further, the story of the statue named "Tecumseh" at the US Naval Academy, installed in 1866 at the Academy, is instructive. The statue was actually of another Native American, Tamanend, and served as the bowsprit of the Delaware, sunk in the Civil War. It ended up being named Tecumseh instead, which indicates with some clarity that (a) the name was well known, and (b) the name was respected by the midshipmen of the time. Respecting the qualities of one's enemies is not an unusual occurrence, particularly after several generations have passed.
Now, why was Tecumseh still known some 50 years after his death? An article in the Indiana Magazine of History from 1989 sheds some background. It would appear that his life was written about quite often.
The first body of literature, coming from a predominantly antebellum romantic school, portrayed Tecumseh as the noble savage. This forceful interpretation persists in popular and academic writings to the present day even though it faded after the Civil War.
Further, the article has evidence that he was written about with much praise shortly after his death. It quotes an 1820 letter to the Indiana Centinal [sic]:
Every schoolboy in the Union now knows that Tecumseh was a great man. He was truly great - and his greatness was his own, unassisted by science or the aid of education. As a statesman, a warrior and a patriot, take him all in all, we shall not look upon his like again.
Combined with multiple pre Civil War biographies of Tecumseh, it is clear that the story of Tecumseh was broadly known and often romanticized. It would seem quite likely that, when the US Navy went in search of names for a class of ships named after Native Americans, Tecumseh's name would be high on the list.
As for the names in the Canonicus (and other monitors) class, the Monitor Center notes:
A note on the names of these vessels seems in order. Secretary of the Navy Gideon Welles directed that new vessels being built should illustrate the pride of the American nation by having distinctly American names. As a result, many of the monitors received names of American rivers, lakes, mountains, cities or Indian tribes. This practice created a list of names that in some cases proved nearly unpronounceable. The practice nevertheless remained in place until 1869, when the new Secretary of the Navy, Adolph A. Borie, ordered the wholesale renaming of ships, often adopting new names based on classical Greek figures or gods. This practice has somewhat complicated for many the tracing of these Civil War era ships.