common pet form of the fem. proper name Margaret, attested since Middle English. Compare magpie. Mag's tales "far-fetched stories, nonsense" is from early 15c.
fem. proper name, an assibilated form of Mag, pet form of Margaret. Also used as the name of a barn-owl and a magpie.
MADGE. The private parts of a woman. [Grose, "Dictionary of the Vulgar Tongue," London, 1785]