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).
This folk etymology was canonized by Aeschylus in Prometheus Bound (v. 734f.), where Prometheus prophesies to Io that the strait would be named after her. 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-, Bosphorus 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 Bosphorus, 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.