"unmarried woman," 1896, from bachelor with French ending -ette. Replaced earlier bachelor-girl (1888). The word appears to have been formed in English; Old French had bachelette "young girl" (15c.), also bachelle, bacelette, bachelote; Modern French is said to use bachelière only in the "student" sense.
Jessica. Thanks! Is Mr. Sparrow a bachelor?
Miss Cornelia. He is, and in all probability will remain one.
Jessica. I'm beginning to think there are pleasanter conditions in this world than being a bachelorette.
Miss Cornelia. Where did you ever pick up such a remarkable word? A bachelorette! What in the world is a bachelorette?
Jessica (smiling). You are.
["The Dummy," Alice Yates Grant, 1896]