1560s, "one who or that which flaps," agent noun from flap (v.). Sense of "forward young woman" is 1921 slang, but the exact connection is disputed. Perhaps from flapper "young wild-duck or partridge" (1747), with reference to flapping wings while learning to fly, many late 19c. examples of which are listed in Wright's "English Dialect Dictionary" (1900), including one that defines it as "A young partridge unable to fly. Applied in joke to a girl of the bread-and-butter age."
Other suggested sources include a late 19c. northern English dialectal use of the word for "teen-age girl" (on notion of one with the hair not yet put up), or an earlier meaning "prostitute" (1889), which is perhaps from dialectal flap "young woman of loose character" (1610s). Any or all of these might have converged in the 1920s sense. Wright also has flappy, of persons, "wild, unsteady, flighty," with the note that it also was "Applied to a person's character, as 'a flappy lass,'" and further on he lists flappy sket (n.) "an immoral woman." In Britain the word took on political tones in reference to the debate over voting rights.
"Flapper" is the popular press catch-word for an adult woman worker, aged twenty-one to thirty, when it is a question of giving her the vote under the same conditions as men of the same age. ["Punch," Nov. 30, 1927]