Obviously, there's a region in Asia Minor that itself comes from the Greek pontos, specifically Pontus. Someone hailing from there could be called ponticus, as shown by Tertullian:
And by a pseudonymous third century text likely based on an earlier Greek anti-heretical tract:
(Similarly, for example, Cicero refers frequently to Heraclides Ponticus, Heraclides from Pontus. Diogenes Laertius refers to the same as Ἡρακλείδης Ποντικὸς, in Greek.)
Ponticus could also be a name, and as a name it could be confused with Pontius in manuscripts. Notice the footnote: "the same ms. reads Ponticus [instead of Pontius] with most mss. of Eusebius."
So, to put a hypothesis out there plainly, I would like to suggest that the stereotyped phrase "before Pontius Pilate" in connection with Jesus' suffering and death found in second century writing connected with the Roman church had a certain propaganda value. Without actually naming the target and giving him a place of dignity in their professions of faith, the phrase would have a chilling effect on the listener when understood as a condemnation of him who is not named. The hypothesis cannot really rise above a conjecture, but it has the advantage of being the only explanation to date as to why it is such a frozen phrase. That is, other than the old chestnut that it was used to lend specificity to the historical setting. (Although not perfect, there may be a modern analogy of sorts.)
Outside of the New Testament (and a fragment of a quotation of 1 Tm 6:13 by Clement of Alexandria), there are three authors in Greek during the second century who use the phrase ἐπὶ Ποντίου Πιλάτου: they are the author of the Ignatian epistles, Justin Martyr, and Irenaeus of Lyons. The first prolific Latin author, Tertullian, also brandishes the Latin sub Pontio Pilato.
Taking them by order of frequency:
In addition to a quotation of 1 Tm 6:13, Tertullian speaks of an unchanging rule of faith:
In this form, it's clearly reminiscent of the creeds. It also has apparent value in refuting heretics, as it is claimed to be immutable.
Irenaeus writes:
There's only a phrase snatched out of a confession of faith here, inserted into a discussion of Jesus' works not being in appearance only, but here:
Irenaeus provides a full expression of the "ancient tradition" which all could follow, in the absence of texts.
Here is the opinion of Irenaeus on Marcion:
There are also two appearances of the phrase in the recensio media of Ignatius, which is of uncertain date and authorship.
I give glory to Jesus Christ the God who bestowed such wisdom upon you; for I have perceived that ye are established in faith immovable, being as it were nailed on the cross of the Lord Jesus Christ, in flesh and in spirit, and firmly grounded in love in the blood of Christ, fully persuaded as touching our Lord that He is truly of the race of David according to the flesh, but Son of God by the Divine will and power, truly born of a virgin and baptized by John that all righteousness might be fulfilled by Him, 1:2 truly nailed up in the flesh for our sakes under Pontius Pilate and Herod the tetrarch [ἀληθῶς ἐπὶ Ποντίου Πιλάτου καὶ Ἡρώδου τετράρχου καθηλωμένον ὑπὲρ ἡμῶν ἐν σαρκί] (of which fruit are we -- that is, of His most blessed passion); that He might set up an ensign unto all the ages through His resurrection, for His saints and faithful people, whether among Jews or among Gentiles, in one body of His Church. (Smyrnaeans 1:1-2)
But the most frequent user of the phrase is Justin Martyr, that most contemporary and bitter opponent of Marcion.
Chapter 13, section 3, line 3
Our teacher of these things is Jesus Christ, who also was born for this purpose, and was crucified under Pontius Pilate [τὸν σταυρωθέντα ἐπὶ Ποντίου Πιλάτου], procurator of Judaea, in the times of Tiberius Caesar; and that we reasonably worship Him, having learned that He is the Son of the true God Himself, and holding Him in the second place, and the prophetic Spirit in the third, we will prove. For they proclaim our madness to consist in this, that we give to a crucified man a place second to the unchangeable and eternal God, the Creator of all; for they do not discern the mystery that is herein, to which, as we make it plain to you, we pray you to give heed.
Justinus Martyr Apol., Apologia
Chapter 35, section 9, line 1
And after He was crucified they cast lots upon His vesture, and they that crucified Him parted it among them. And that these things did happen, you can ascertain from the Acts of Pontius Pilate. [δύνασθε μαθεῖν ἐκ τῶν ἐπὶ Ποντίου Πιλάτου γενομένων ἄκτων]
Justinus Martyr Apol., Apologia
Chapter 46, section 1, line 5
But lest some should, without reason, and for the perversion of what we teach, maintain that we say that Christ was born one hundred and fifty years ago under Cyrenius, and subsequently, in the time of Pontius Pilate [δεδιδαχέναι δὲ ἅ φαμεν διδάξαι αὐτὸν ὕστερον χρόνοις ἐπὶ Ποντίου Πιλάτου], taught what we say He taught; and should cry out against us as though all men who were born before Him were irresponsible--let us anticipate and solve the difficulty. We have been taught that Christ is the first-born of God, and we have declared above that He is the Word of whom every race of men were partakers; and those who lived reasonably are Christians, even though they have been thought atheists; as, among the Greeks, Socrates and Heraclitus, and men like them; and among the barbarians, Abraham, and Ananias, and Azarias, and Misael, and Elias, and many others whose actions and names we now decline to recount, because we know it would be tedious. So that even they who lived before Christ, and lived without reason, were wicked and hostile to Christ, and slew those who lived reasonably. But who, through the power of the Word, according to the will of God the Father and Lord of all, He was born of a virgin as a man, and was named Jesus, and was crucified, and died, and rose again, and ascended into heaven, an intelligent man will be able to comprehend from what has been already so largely said. And we, since the proof of this subject is less needful now, will pass for the present to the proof of those things which are urgent.
Justinus Martyr Apol., Apologia
Chapter 48, section 3, line 2
And that it was predicted that our Christ should heal all diseases and raise the dead, hear what was said. There are these words: "At His coming the lame shall leap as an hart, and the tongue of the stammerer shall be clear speaking: the blind shall see, and the lepers shall be cleansed; and the dead shall rise, and walk about." And that He did those things, you can learn from the Acts of Pontius Pilate. [ἐκ τῶν ἐπὶ Ποντίου Πιλάτου γενομένων ἄκτων μαθεῖν δύνασθε]
Justinus Martyr Apol., Apologia
Chapter 61, section 13, line 2
And for this [rite] we have learned from the apostles this reason. Since at our birth we were born without our own knowledge or choice, by our parents coming together, and were brought up in bad habits and wicked training; in order that we may not remain the children of necessity and of ignorance, but may become the children of choice and knowledge, and may obtain in the water the remission of sins formerly committed, there is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe; he who leads to the layer the person that is to be washed calling him by this name alone. For no one can utter the name of the ineffable God; and if any one dare to say that there is a name, he raves with a hopeless madness. And this washing is called illumination, because they who learn these things are illuminated in their understandings. And in the name of Jesus Christ, who was crucified under Pontius Pilate [τοῦ σταυρωθέντος ἐπὶ Ποντίου Πιλάτου], and in the name of the Holy Ghost, who through the prophets foretold all things about Jesus, he who is illuminated is washed.
Justinus Martyr Apol., Apologia secunda
Chapter 6, section 6, line 4
But "Jesus," His name as man and Saviour, has also significance. For He was made man also, as we before said, having been conceived according to the will of God the Father, for the sake of believing men, and for the destruction of the demons. And now you can learn this from what is under your own observation. For numberless demoniacs throughout the whole world, and in your city, many of our Christian men exorcising them in the name of Jesus Christ, who was crucified under Pontius Pilate [τοῦ σταυρωθέντος ἐπὶ Ποντίου Πιλάτου], have healed and do heal, rendering helpless and driving the possessing devils out of the men, though they could not be cured by all the other exorcists, and those who used incantations and drugs.
Justinus Martyr Apol., Dialogus cum Tryphone
Chapter 30, section 3, line 8
Moreover, it is also manifest to all, that we who believe in Him pray to be kept by Him from strange, i.e., from wicked and deceitful, spirits; as the word of prophecy, personating one of those who believe in Him, figuratively declares. For we do continually beseech God by Jesus Christ to preserve us from the demons which are hostile to the worship of God, and whom we of old time served, in order that, after our conversion by Him to God, we may be blameless. For we call Him Helper and Redeemer, the power of whose name even the demons do fear; and at this day, when they are exorcised in the name of Jesus Christ, crucified under Pontius Pilate [τοῦ σταυρωθέντος ἐπὶ Ποντίου Πιλάτου], governor of Judaea, they are overcome. And thus it is manifest to all, that His Father has given Him so great power, by virtue of which demons are subdued to His name, and to the dispensation of His suffering.
Justinus Martyr Apol., Dialogus cum Tryphone
Chapter 76, section 6, line 4
And again, in other words, He said, 'I give unto you power to tread on serpents, and on scorpions, and on scolopendras, and on all the might of the enemy.' And now we, who believe on our Lord Jesus, who was crucified under Pontius Pilate [οἱ πιστεύοντες ἐπὶ τὸν σταυρωθέντα ἐπὶ Ποντίου Πιλάτου Ἰησοῦν κύριον ἡμῶν], when we exorcise all demons and evil spirits, have them subjected to us. For if the prophets declared obscurely that Christ would suffer, and thereafter be Lord of all, yet that [declaration] could not be understood by any man until He Himself persuaded the apostles that such statements were expressly related in the Scriptures. For He exclaimed before His crucifixion: 'The Son of man must suffer many things, and be rejected by the Scribes and Pharisees, and be crucified, and on the third day rise again.' And David predicted that He would be born from the womb before sun and moon, according to the Father's will, and made Him known, being Christ, as God strong and to be worshipped.
Justinus Martyr Apol., Dialogus cum Tryphone
Chapter 85, section 2, line 4
For every demon, when exorcised in the name of this very Son of God--who is the First-born of every creature, who became man by the Virgin, who suffered, and was crucified under Pontius Pilate by your nation [καὶ σταυρωθέντος ἐπὶ Ποντίου Πιλάτου ὑπὸ τοῦ λαοῦ ὑμῶν καὶ ἀποθανόντος], who died, who rose from the dead, and ascended into heaven--is overcome and subdued.
Apology 61 most clearly suggests the use of the phrase in baptism (which in turn connects it to the creeds), and Dialogue 85 has the familiar sequence of crucifixion under Pontius Pilate, death, resurrection, and ascension.
And here is Justin's opinion of Marcion:
And, as we said before, the devils put forward Marcion of Pontus [Μαρκίωνα δὲ τὸν ἀπὸ Πόντου], who is even now teaching men to deny that God is the maker of all things in heaven and on earth, and that the Christ predicted by the prophets is His Son, and preaches another god besides the Creator of all, and likewise another son. And this man many have believed, as if he alone knew the truth, and laugh at us, though they have no proof of what they say, but are carried away irrationally as lambs by a wolf, and become the prey of atheistical doctrines, and of devils. For they who are called devils attempt nothing else than to seduce men from God who made them, and from Christ His first-begotten; and those who are unable to raise themselves above the earth they have riveted, and do now rivet, to things earthly, and to the works of their own hands; but those who devote themselves to the contemplation of things divine, they secretly beat back; and if they have not a wise sober-mindedness, and a pure and passionless life, they drive them into godlessness. (First Apology, 58)
Last but not least, in the New Testament, the only appearances of "Pontius" Pilate are at Luke 3:1 (cf. Dieter Roth), Acts 4:27, and 1 Timothy 6:13. Both the present form of Luke-Acts and the pastoral epistles are believed by some scholars to harbor anti-Marcionite sentiment, on independent grounds.