ancient king of Phrygia, 1560s; the name is of Phrygian origin. He was given by the gods the gift of turning all he touched to gold, but as this included his food he had to beg them to take it back again. Hence Midas touch (1883). But the oldest references to him in English are to the unrelated story of the ass's ears given him by Apollo for being dull to the charms of his lyre.
c. 1500, kind of small bird that has its tail in continuous motion (late 12c. as a surname), earlier wagstart (mid-15c.), from wag (v.) + tail (n.). From 18c. as "a harlot," but this sense seems to be implied much earlier:
If therefore thou make not thy mistress a goldfinch, thou mayst chance to find her a wagtaile. [Lyly, "Midas," 1592]