Where did the nomenclature ”godfather” originate from?
A godfather is any man who serves as a sponsor for a child at baptism.
Godfather (n.) [is a] man who sponsors one at baptism and guarantees the child's religious education, late 12c., from God + father (n.), modifying or replacing Old English godfaeder.
Thus it is safe to assume that you are not talking about the the movie The Godfather or the cocktail named The Godfather. Of course a godfather could have a ”Godfather” after the baptism of his newly baptized godson or goddaughter.
For the sake of clarity I will deal more with the term ”godparents” than either ”godfather” or ”godmother”.
A godparent in infant baptism practicing denominations of Christianity, is someone who bears witness to a child's christening and later is willing to help in their catechesis, as well as their lifelong spiritual formation. In the past, in some countries, the role carried some legal obligations as well as religious responsibilities. In both religious and civil views, a godparent tends to be an individual chosen by the parents to take an interest in the child's upbringing and personal development, to offer mentorship or claim legal guardianship of the child if anything should happen to the parents. A male godparent is a godfather, and a female godparent is a godmother. The child is a godchild (i.e. godson for boys and goddaughter for girls).
Origins and history
As early as the 2nd century AD, infant baptism had begun to gain acceptance among Catholic Christians for the spiritual purification and social initiation of infants,nthe requirement for some confession of faith necessitated the use of adults who acted as sponsors for the child. They vocalized the confession of faith and acted as guarantors of the child's spiritual beliefs.
Normally, these sponsors were the natural parents of a child, as emphasized in 408 by St. Augustine who suggested that they could, it seems exceptionally, be other individuals.5 Within a century, the Corpus Juris Civilis indicates that parents had been replaced in this role almost completely.5 This was clarified in 813 when the Synod of Mainz prohibited natural parents from acting as godparents to their own children.5
By the 5th century, male sponsors were referred to as "spiritual fathers", and by the end of the 6th century, they were being referred to as "compaters" and "commaters", suggesting that these were being seen as spiritual co-parents. This pattern was marked by the creation of legal barriers to marriage that paralleled those for other forms of kin. A decree of Justinian, dated to 530, outlawed marriage between a godfather and his goddaughter, and these barriers continued to multiply until the 11th century, forbidding marriage between natural and spiritual parents, or those directly related to them. As confirmation emerged as a separate rite from baptism from the 8th century, a second set of sponsors, with similar prohibitions, also emerged. The exact extent of these spiritual relationships as a bar to marriage in Catholicism was unclear until the Council of Trent, which limited it to relationships between the godparents, the child, and the parents.
[1.] W. Parsons, ed., Saint Augustine, Letters, The Fathers of the Church, 18 (New York, 1953), pp. 134-5.
[2.] P. Kruger, ed., Corpus Iuris Civis, vol. 3, Codex Iustinianus (Dublin and Zurich, 1970), v, 4, 26, p. 197.
The Catholic Encyclopedia explain# the term of sponsor as follows:
When infants are solemnly baptized, persons assist at the ceremony to make profession of the faith in the child's name. This practice comes from antiquity and is witnessed to by Tertullian, St. Basil, St. Augustine, and others. Such persons are designated sponsores, offerentes, susceptores, fidejussores, and patrini. The English term is godfather and godmother, or in Anglo-Saxon, gossip.
These sponsors, in default of the child's parents, are obliged to instruct it concerning faith and morals. One sponsor is sufficient and not more than two are allowed. In the latter case, one should be male and the other female. The object of these restrictions is the fact that the sponsor contracts a spiritual relationship to the child and its parents which would be an impediment to marriage. Sponsors must themselves be baptized persons having the use of reason and they must have been designated as sponsors by the priest or parents. During the baptism they must physically touch the child either personally or by proxy. They are required, moreover, to have the intention of really assuming the obligations of godparents. It is desirable that they should have been confirmed, but this is not absolutely necessary. Certain persons are prohibited from acting as sponsors. They are: members of religious orders, married persons in respect to each other, or parents to their children, and in general those who are objectionable on such grounds as infidelity, heresy, excommunication, or who are members of condemned secret societies, or public sinners (Sabetti, no. 663). Sponsors are also used in the solemn baptism of adults. They are never necessary in private baptism.
The role of godparents really became commonplace around 800 AD.
In the early centuries after Jesus’ death and Resurrection (about 300 A.D.), “the Church was under the persecution of the Roman Empire and had to be cautious in conducting its affairs so as to prevent pagan infiltration and persecution” (Fr. William Saunders). In addition, during Medieval times, the Sacraments of Initiation were administered concurrently (e.g., Baptism, Eucharist, and Confirmation), so the role of Godparents really was twofold. First, Godparents were essential in attesting to the integrity of the individual who was joining the Catholic faith, often because s/he was an adult receiving the Sacraments of Initiation (as in RCIA today). Second, Godparents had to protect the doctrine of the Faith from paganism and persecution in the early days. Finally, Godparents were critical spiritual guides if parents of a child were martyred and had no direction in the Faith.
Around the year 800 A.D., infant Baptism became commonplace, and it was then that the role of Godparent (or more accurately “sponsor”) significantly changed to what we understand it to be today. Sponsors were intended to be adults who committed themselves to assist parents of children to teach them the Catholic Faith. - History of Godparents In The Catholic Church
For more information one can peruse the following articles: