When the time arrived for Elizabeth to have her child she gave birth to a son. Her neighbors and relatives heard that the Lord had shown his great mercy toward her, and they rejoiced with her. When they came on the eighth day to circumcise the child, they were going to call him Zechariah after his father, but his mother said in reply, "No. He will be called John." But they answered her, "There is no one among your relatives who has this name." So they made signs, asking his father what he wished him to be called. He asked for a tablet and wrote, "John is his name," and all were amazed. Immediately his mouth was opened, his tongue freed, and he spoke blessing God. Then fear came upon all their neighbors, and all these matters were discussed throughout the hill country of Judea. All who heard these things took them to heart, saying, "What, then, will this child be?" For surely the hand of the Lord was with him. The child grew and became strong in spirit, and he was in the desert until the day of his manifestation to Israel. (Lk 1:57-66,80)
Birth of a child is due to the mercy of God. Relatives and neighbours recognized this and they rejoiced over this. Yes, birth of a child is a time of gathering of relatives and friends. Also it is a time to rejoice and celebrate. It is also a time to remember God and be grateful for the mercy shown. Every birth reminds us of unity (gathering of relatives and friends), makes us celebrate and helps us to remind us the presence of God and his mercy.
Naming of the child shows us that the child belongs to the family. Why? Because, they always named the child with the name of the father or with the name of the forefathers of the family. So it was to show the family to which the child belonged. But here the name is revealed by God and that shows that he belonged to God and the child has a mission that is given by God.
It is strange that the child grew up but was in the desert till the day of the manifestation. We know that the desert is the place of God experience. (Israel experienced God in the desert while they were there for 40 years). So John who was a gift of God (by birth) belonged to God (by his name) and grew up with God (in the desert). All these point out to him that he has a special mission.