Heidegger's Bible Handbook: 3 John: Recipient
2. It is inscribed to Gaius, or Caius, not a Corinthian, but perhaps a Roman.
It is inscribed to Γαΐῳ τῷ ἀγαπητῷ, the wellbeloved Gaius, verse 1. Whether or not this is the Gaius of whom mention is made in Romans 16:23; 1 Corinthians 1:14, and who was a Corinthian, and converted by Saint Paul, is uncertain. But the reckoning of the time will hardly bear this. And, that this man was concerted by Saint John, is demonstrated out of verse 4. But the former appears to have been convered by Saint Paul. That he was a Roman or Latin, the name Gaius, or Caius, appears to indicate.