How did the mute Zechariah make his son's name known?

Answer: He wrote it on a tablet

Zechariah asked for a tablet and wrote, "His name is John." Immediately his mouth was opened and he praised God.

“So they made signs to his father to find out what he wanted to name the child. Zechariah asked for a tablet and wrote, “His name is John.” And they were all amazed. Immediately Zechariah’s mouth was opened and his tongue was released, and he began to speak, praising God.”— Luke 1:62-64 (BSB)