1961, abbreviation of mini-car, a small car made by British Leyland (formerly British Motor Corp.). As an abbreviation of miniskirt, it is attested from 1966.
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miniskirt n.
also mini-skirt, "skirt with a hem-line well above the knee," 1965, from mini- + skirt (n.); reputedly the invention of French fashion designer André Courrèges (1923-2016).
"The miniskirt enables young ladies to run faster, and because of it, they may have to." [John V. Lindsay, New York Times, Jan. 13, 1967]