In Luke 1:59-60 we also have the account of the birth of John the Baptist:
And it came to pass, that on the eighth day they came to circumcise the child; and they called him Zacharias, after the name of his father. And his mother answered and said, Not so; but he shall be called John.
So at the time Luke was writing, naming a boy at his circumcision seems to be a general practice, but was it a requirement of the Law?
There is no specific mention of such a requirement in the covenant instructions given by God to Abraham in Genesis 17:9-14.
Genesis 21:1-4 records the birth of Isaac:
And the LORD visited Sarah as he had said, and the LORD did unto Sarah as he had spoken. For Sarah conceived, and bare Abraham a son in his old age, at the set time of which God had spoken to him. And Abraham called the name of his son that was born unto him, whom Sarah bare to him, Isaac. And Abraham circumcised his son Isaac being eight days old, as God had commanded him.
Which gives the impression that Abraham named Isaac prior to his circumcision.
The practice of adult circumcision during Abraham's time, Moses' wilderness journey (Exodus 12:48), Joshua's entry into Canaan (Joshua 5:1-9), and that required by the Jewish Christians (Acts 15), are a clear indication that circumcision as a religious requirement was not connected to the naming of male children.
In John 7, Jesus responds to the Jewish leaders' astonishment at his authority and knowledge:
Moses therefore gave unto you circumcision; (not because it is of Moses, but of the fathers;) and ye on the sabbath day circumcise a man. If a man on the sabbath day receive circumcision, that the law of Moses should not be broken; are ye angry at me, because I have made a man every whit whole on the sabbath day? Judge not according to the appearance, but judge righteous judgment.
His primary purpose here was to challenge the Jewish leaders in regard to the letter of the law versus the spirit of the law, but his aside is interesting also. Jesus is indicating here that the Jewish practice of circumcision had become disconnected from the Abrahamic Covenant and was now merely a tradition handed to them by their fathers - the significance of it had been lost altogether.
Jesus drew a clear distinction between the religious requirements of the Law that was given to Moses, and the religious requirements of tradition that had been added to the Law over time. The religious requirements of tradition were purely cultural and a significant source of concern for Jesus. Mark 7:5-9 records:
Then the Pharisees and scribes asked him, Why walk not thy disciples according to the tradition of the elders, but eat bread with unwashen hands?
He answered and said unto them, Well hath Esaias prophesied of you hypocrites, as it is written, This people honoureth me with their lips, but their heart is far from me. Howbeit in vain do they worship me, teaching for doctrines the commandments of men. For laying aside the commandment of God, ye hold the tradition of men, as the washing of pots and cups: and many other such like things ye do. And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition.
There is nothing in the commandments given to Abraham or Moses in relation to circumcision that requires the naming of a male child at the time of his circumcision. So, the practice as recorded by Luke is simply cultural, being part of Jewish tradition and hence not required in order to maintain a person's right relationship with God.