Luke 1:57-66 (ESV)
Now the time came for Elizabeth to give birth, and she bore a son.  And her neighbors and relatives heard that the Lord had shown great mercy to her, and they rejoiced with her.  And on the eighth day they came to circumcise the child. And they would have called him Zechariah after his father,  but his mother answered, "No; he shall be called John."  And they said to her, "None of your relatives is called by this name."  And they made signs to his father, inquiring what he wanted him to be called.  And he asked for a writing tablet and wrote, "His name is John." And they all wondered.  And immediately his mouth was opened and his tongue loosed, and he spoke, blessing God.  And fear came on all their neighbors. And all these things were talked about through all the hill country of Judea,  and all who heard them laid them up in their hearts, saying, "What then will this child be?" For the hand of the Lord was with him.
Luke is the most detailed of the gospels. One of the details is the circumcisions he records for both John the Baptist, or Baptizer, and Jesus Christ. The other gospels don’t record these. Most of the gospels give almost no attention to the Birth of John the Baptizer. But Luke does. One does wonder where he gets these details, as John was dead, and you would think as old as his parents were when he was born they were dead by then too. Perhaps he gets the info from Mary just as he gets the info concerning her three month visit? If Elizabeth was six months pregnant when Mary was notified of it, then Maries three month visit would have concluded after Elizabeth had given birth. She would have been an eyewitness to these things and the best one to report them. Of course this places the writing of Luke very early, but I don’t have a problem with that.
Zecharia was dumb during this period but could write instructions. He says the child’s name is John. Only after this can he begin to speak. This alarms the neighbors, and they expect the child to do great things. He does not fail to deliver. I do wonder if we would be smart enough today to see the connection revolving around a child’s birth such as this, and know that this would mean great things to come? Probably, we would write off such notions as superstition. But the neighbors knew that this meant the hand of the Lord was with him.