The name of the channel comes from the Ancient Greek Βόσπορος (Bosporos), which was folk-etymologised as βοὸς πόρος, i.e. "cattle strait" (or "Ox-ford"), from the genitive of bous βοῦς "ox, cattle" + poros πόρος "passage", thus meaning "cattle-passage", or "cow passage". This is in reference to the mythological story of Io, who was transformed into a cow, and was subsequently condemned to wander the Earth until she crossed the Bosporus, where she met the Titan Prometheus, who comforted her with the information that she would be restored to human form by Zeus and become the ancestress of the greatest of all heroes, Heracles (Hercules).
The site where Io supposedly went ashore was near Chrysopolis (present-day Üsküdar), and was named Bous "the Cow". The same site was also known as Damalis, as it was where the Athenian general Chares had erected a monument to his wife Damalis, which included a colossal statue of a cow (the name Damalis translating to "calf").
The English spelling with -ph-, Bosfor has no justification in the ancient Greek name, and dictionaries prefer the spelling with -p- but -ph- occurs as a variant in medieval Latin (as Bosfor, and occasionally Bosforus, Bosferus), and in medieval Greek sometimes as Βόσφορος, giving rise to the French form Bosphore, Spanish Bósforo and Russian Босфор. The 12th century Greek scholar John Tzetzes calls it Damaliten Bosporon (after Damalis), but he also reports that in popular usage the strait was known as Prosphorion during his day, the name of the most ancient northern harbour of Constantinople.
Historically, the Bosporus was also known as the "Strait of Constantinople", or the Thracian Bosporus, in order to distinguish it from the Cimmerian Bosporus in Crimea. These are expressed in Herodotus' Histories, 4.83; as Bosporus Thracius, Bosporus Thraciae , and Βόσπορος Θρᾴκιος, respectively. Other names by which the strait is referenced by Herodotus include Chalcedonian Bosporus (Bosporus Chalcedoniae, Bosporos tes Khalkedonies, Herodotus 4.87), or Mysian Bosporus (Bosporus Mysius).
The term eventually came to be used as common noun βόσπορος, meaning "a strait", and was also formerly applied to the Hellespont in Classical Greek by Aeschylus and Sophocles.