Succession lists of kings, periodically appointed magistrates, and heads of philosophical schools were kept in the Hellenistic world. The Jews had lists of prophets and rabbis, but most importantly of high priests. Although early Christians had an interest in the succession of their own prophets and teachers (particularly in the catechetical school in Alexandria), special attention attached to the succession of bishops, who by the end of the second century incorporated much of the authority and function of prophets and teachers into their office.
1 Clement 42-44 taught the apostolic institution of the offices of bishop and deacon in the church. After the appointment of the first bishops and deacons, the apostles provided for the continuation of these offices in the church. This was not the same as the later doctrine of apostolic succession, and it is to be noted that Clement included deacons as well as bishops in his statement. Ignatius, the first witness to only one bishop in a church, did not base his understanding of the ministry on succession. The one bishop was a representative of God the Father, and the presbyters had their model in the college of apostles (Trall. 3).
The first claim to a succession from the apostles in support of particular doctrines was made in the second century by the Gnostics. They claimed that the apostles had imparted certain secret teachings to some of their disciples and that these teachings had been passed down, thus having apostolic authority, even if different from what was proclaimed in the churches (Irenaeus, Haer. 3.2.1; cf. Ptolemy in Epiphanius, Haer. 33.7.9). Hegesippus, an opponent of Gnosticism, compiled a list of the bishops in Rome (Eusebius, H.E. 4.22.5f.).
Irenaeus of Lyons drew on the idea of the succession of bishops to formulate an orthodox response to the Gnostic claim of a secret tradition going back to the apostles. Irenaeus argued that if the apostles had had any secrets to teach, they would have delivered them to those men to whom they committed the leadership of the churches. A person could go to the churches founded by apostles, Irenaeus contended, and determine what was taught in those churches by the succession of teachers since the days of the apostles. The constancy of this teaching was guaranteed by its public nature; any change could have been detected, since the teaching was open. The accuracy of the teaching in each church was confirmed by its agreement with what was taught in other churches. One and the same faith had been taught in all the churches since the time of the apostles.
Irenaeus's succession was collective rather than individual. He spoke of the succession of the presbyters (Haer. 3.2.2), or of the presbyters and bishops (4.26.2), as well as of the bishops (3.3.1). To be in the succession was not itself sufficient to guarantee correct doctrine. The succession functioned negatively to mark off the heretics who withdrew from the church. A holy life and sound teaching were also required of true leaders (4.26.5). The succession pertained to faith and life rather than to the transmission of special gifts. The "gift of truth" (charisma veritatis) received with the office of teaching (4.26.2) was not a gift guaranteeing that what was taught would be true, but was the truth itself as a gift. Each holder of the teaching chair in the church received the apostolic doctrine as a deposit to be faithfully transmitted to the church. Apostolic succession as formulated by Irenaeus was from one holder of the teaching chair in a church to the next and not from ordainer to ordained, as it became....
[In Tertullian] Churches were apostolic that agreed in the same faith, even if not founded by apostles.
Apostolic succession arose in a polemical situation as an effective argument for the truth of Catholic tradition against Gnostic teachings. As so often happens to successful arguments, it came to be regarded as an article of faith, not just a defense of the truth but a part of truth itself.
Hippolytus is apparently the first for whom the bishops were not simply in the succession from the apostles but were themselves successors of the apostles (Haer., praef.). When Eusebius of Caesarea used the lists of bishops as the framework for his Church History, he did not count the apostles in the episcopal lists. Cyprian, however, made an identification of the episcopate and the apostolate (Ep. 64.3; 66.4; cf. Sent. epp. 79 and Socrates, H.E. 6.8)....
The sacramental understanding of ordination that grew up in the fourth and fifth centuries shifted the emphasis to a succession from ordainer to ordained, but the earlier historical type of succession was preserved in the lists of local bishops....
Election by the people was one of the methods of appointment known to Origen (Hom. 13 in Num. 4)....
The will of the populace could prevail over clerical opposition (Sulpicius Severus, V. Mart. 9). (Encyclopedia Of Early Christianity [New York: Garland Publishing, Inc., 1999], pp. 94-95, 366-367)
Robert Lee Williams did his doctoral dissertation on a subject related to apostolic succession in 1982, under Robert Grant at the University of Chicago. He recently published a revised version of his dissertation, under the title Bishop Lists (Piscataway, New Jersey: Gorgias Press, 2005). He reaches some of the same conclusions outlined by Ferguson above. He notes that there were many concepts of succession circulating in the ancient world, in political circles, among philosophers, etc. Different individuals and groups defined succession in different ways, and the early Christians adopted multiple concepts of succession in multiple contexts. Williams notes, for example, that Eusebius' church history begins, "It is my purpose to write an account of the successions of the holy apostles" (1:1), referring not only to successions of bishops, but also to successions of Christian philosophers, for example (pp. 201, 203, 211, 220). Williams argues that Eusebius was largely influenced by the Jewish historian Josephus in how he formulated his ideas related to succession (pp. 213-214).
The popularity of succession concepts in antiquity is important for more than one reason. It's a possible explanation of where concepts of apostolic succession came from if they weren't taught by the apostles. And the widespread use and development of succession concepts suggests that the early Christians, including the New Testament authors, could have explicitly referred to such ideas if they had wanted to. The absence of evidence for apostolic succession in the earliest Christian sources can't be explained by an alleged unfamiliarity with such concepts.
Remember, when Dave Armstrong was explaining his dubious claim that "we find an explicit espousal of apostolic succession" in Papias, he wrote, "the word 'explicit' was relative insofar as someone that early can only be so explicit. 'Direct' would have been a better term to use in retrospect, because of the meaning of 'explicit' in discussions having to do with development of doctrine". Elsewhere, he wrote, "Jason's mistake is what I noted early on: he irrationally expects to find the full-blown oak tree when it is only reasonable to find the acorn or small tree." But the more the early Christians were familiar with concepts of succession, and the more relevant terminology they had access to, the less likely it becomes that we should expect their alleged belief in Dave's notion of apostolic succession to only be expressed in seed form and in ways that are so easily denied by Protestant scholars and others.