In Greek mythology, Myconus (, also ; Ancient Greek: Μύκονος Mykonos) or Mycon was a local hero and an eponymous first ruler of Mykonos.Stephanus of Byzantium s.v. Andros & Mykonos He was the son or grandson of the god Apollo.
Mythology Mycon was the father of Xanthippe who fed him with her own breastmilk to prevent him from dying of starvation when he was imprisoned.Hyginus, Fabulae 254 The daughter was also known as Pero.Valerius Maximus, 5.4. ext.1
The island of Mykonos was also said to have been the location of the Gigantomachy, the great battle between Zeus and Giants and where Hercules killed the invincible giants, having lured them from the protection of Mount Olympus. According to myth, the large rocks all over the island are said to be the petrified corpses of the giants.
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References
Gaius Julius Hyginus, Fabulae from The Myths of Hyginus translated and edited by Mary Grant. University of Kansas Publications in Humanistic Studies. Online version at the Topos Text Project. Stephanus of Byzantium, Stephani Byzantii Ethnicorum quae supersunt, edited by August Meineike (1790-1870), published 1849. A few entries from this important ancient handbook of place names have been translated by Brady Kiesling. Online version at the Topos Text Project.