America is the feminine form of the name Americus, which is the Latinized version of the Italian name Amerigo. Amerigo Vespucci, an Italian explorer from the same era as Christopher Columbus, is attributed with discovering that South America was not part of Asia, but a separate continent. (Amerigo signed the Latin form of his name when writing in Latin.)

Martin Waldseemüller’s world map from 1507 is the earliest use of the name. In an accompanying book, the author explains that since Europe and Asia had been named after women, he proposed that the feminine form of Americus be used as well:

"I do not see what right any one would have to object to calling this part after Americus, who discovered it and who is a man of intelligence, Amerige, that is, the Land of Americus, or America: since both Europa and Asia got their names from women"

Source: John W. Hessler's The Naming of America: Martin Waldseemüller’s 1507 World Map and the Cosmographiae Introductio

Interesting side note: The Latin name Americus, the German Heinrich, and the English Henry all have a common root in the Old High German name Haimirich.