PIE*skerの「回す」「回転する」が語源で、charladyの「指輪」と同じ。
Chore, a corruption of char, is an English word, still used in many parts of England, as a char-man, a char-woman; but in America, it is perhaps confined to New England. It signifies small domestic jobs of work, and its place cannot be supplied by any other single word in the language. [Noah Webster, "Dissertations on the English Language," 1789]