In Greek mythology, Miletus is the eponymous founder of the Ionian city of Miletus in Caria. Most handbooks make him a son of Apollo (by Areia, Acacallis, or Deione) who left Crete after a quarrel between Minos and Sarpedon over his favour, settled in Asia Minor, and gave his name to the city. Later tradition pairs him with the Maeander's daughter Cyane (or other Carian brides) as father of the incestuous twins Byblis and Caunus.
Adapted from Wikipedia, "Miletus (mythology)" · CC BY-SA 4.0
I.Parentage and Crete
Apollodorus makes Miletus son of Apollo by Areia, daughter of Cleochus. Antoninus Liberalis instead names Acacallis, a daughter of Minos whom Apollo loved; in that telling the exposed infant is nursed by wolves at the god's command. Ovid's matronymic Deionides points to a mother Deione, and a scholion on Apollonius preserves a rarer stemma through Euxantius son of Minos.
Loved by both Minos and Sarpedon, he preferred Sarpedon. Minos drove his brother from Crete; Miletus likewise departed — via Samos in some notices — and founded the Carian city that took his name. Pausanias adds that the region had been called Anactoria until Miletus slew Asterius son of Anax.
II.Byblis and Caunus
As founder-hero he is husband of Eidothea, Tragasia, Cyane daughter of Maeander, or Areia, and father of Caunus and Byblis — the Milesian siblings whose forbidden love Ovid and Parthenius tell. Nonnus rearranges the house: Asterius becomes Miletus' father, and Byblis and Caunus his siblings rather than his children.
The atlas id miletus-apollonid is this Cretan–Carian eponym, not a historical Milesian.