The Unitarian Universalist church is living proof that there is some sort of naming component in the child dedication/baptism ceremony in traditional Christian churches. How is the Christian name assigned? By the priest or the child's family? What is the difference between the "Christian name" and the "baptismal name" and the "confirmation name"? And what is the significance of each name?
In some cultures, notably the UK but also others, the term 'Christian name' simply means (or used to mean) someone's first name. So David Cameron's Christian name is 'David'. The logic behind this is that babies were typically 'named' at their baptism, which typically occurred shortly after their birth. Going back thirty or forty years you would find official forms that referred to a person's 'Christian name' and 'surname', and the first name might be referred to like that even if the person had not actually been baptised. Nowadays the term has dropped out of favour as it becomes less and less likely that a person will be even nominally Christian. Someone who was baptized as an infant can still reasonably call their first name their 'Christian name'.
At Confirmation, Catholics and Anglicans may select a new 'Christian' name to go with their baptismal names, and adult converts are encouraged to do so. This is sometimes described as a 'Christian name'. It is not normally part of someone's official or legal name unless they explicitly change it.
As far as the use of Christian name in the protestant faith seems to be a carry over from the Jewish tradition of giving the baby a name at circumcision.
Exactly when and how this tradition began I can find no reference in the Bible, Perhaps someone with more knowledge can further explain.
I have been told but cannot verify that it is in connection with the Abrahamic covenant.
Jesus was likewise given his name at his circumcision on the eighth day.
At some point in History, The Church apparently the Holy Roman Catholic Church, began to give children a new name at baptism, and that name has now apparently become an individual's middle name, and was known as a Christian name (or the name of the new Christian). Somehow over the expanse of time the Christian name evolved to be a person's first name, and this was probably because, in the Jewish tradition the Parents gave the child a name at circumcision which served as confirmation of coveting with God.
After the resurrection of Christ the coveting ceremony of circumcision was replaced by Baptism, the naming of the child was then known as Christening the child and thus the use of Christian name.
As for sir names, that is commonly thought to have come about by a process of elimination. As Jesus was called in his village as the son of the carpenter to distinguish him as separate from any other man named Jesus (Joshua) which was a common name in that day. It seems plausible that over the years that *the son of the carpenter * could have been shortened to just Carpenter. that however is conjecture and not based on any evidence of which I am aware.