Here are five important selections for understanding early Christian reflections on the Old Testament canon. It seems, from my brief study, that the first early Christian reference to the “Old Testament” as a collection of books (cf. 2 Cor 3: 14), comes from Melito (ca. AD 170). However, this text is preserved in a later text, so the dating cannot be fully substantiated. If anyone is aware of an earlier or comparatively early citation, please let me know.
For fuller, yet accessible, treatments of these and other texts on this important topic, see E. Earle Ellis, The Old Testament in Early Christianity (Grand Rapids: Baker, 1991); Roger Beckwith, The Old Testament Canon of the New Testament Church and Its Background in Early Judaism (London: SPCK, 1985, repr. Eugene, OR: Wipf & Stock, 2008); and the resources posted by Ed Gallagher.
Ecclesiasticus 1:1 (my translation):
Πολλῶν καὶ μεγάλων ἡμῖν διὰ τοῦ νόμου καὶ τῶν προφητῶν kαὶ τῶν ἄλλων τῶν κατ᾿ αὐτοὺς ἠκολουθηκότων δεδομένων…
Many and great things we have received through the Law and the Prophets and the others which followed them…
Philo, De Vita Contemplativa 25, speaking of the piety of the Therapeutae (my translation):
μηδὲν εἰσκομίζοντες…ἀλλὰ νόμους καὶ λόγια θεσπισθέντα διὰ προφητῶν καὶ ὕμνους καὶ τὰ ἄλλα οἷς ἐπιστήμη καὶ εὐσέβεια συναύξονται καὶ τελειοῦνται.
nor did they carry in…but the Law and the prophesied words of the prophets and hymns and the others, by which knowledge and godliness grow together and have their completeness.
Melito’s canon, as preserved in Eusebius’s Ecclesiastical History 4.26.14ff. (my translation):
ἀνελθὼν οὖν εἰςτὴνἀνατολὴν καὶ ἕως τοῦ τόπου γενόμενος ἔνθα ἐκηρύχθη καὶ ἐπράχθη, καὶ ἀκριβῶς μαθὼν τὰ τῆς παλαιᾶς διαθήκης βιβλία, ὑπο τάξας ἔπεμψά σοι: ὧν ἐστι τὰ ὀνόματα: Μωυσέως πέντε, Γένεσις Ἔξοδος Ἀριθμοὶ Λευιτικὸν Δευτερονόμιον, Ἰησοῦς Ναυῆ, Κριταί, Ῥούθ, Βασιλειῶν τέσσαρα, Παραλειπομένων δύο, Ψαλμῶν Δαυίδ, Σολομῶνος Παροιμίαιἡκαὶ Σοφία, Ἐκκλησιαστής, Ἆισμα Ἀισμάτων, Ἰώβ, Προφητῶν Ἡσαΐου Ἱερεμίου τῶν δώδεκα ἐν μονοβίβλῳ Δανιήλ, Ἰεζεκιήλ, Ἔσδρας
Therefore, coming to the East, reaching even the place where these things were preached and carried out, and having learned exactly the books of the Old Testament–putting them down, I sent them to you. The names are: the Five of Moses–Genesis, Exodus, Numbers, Leviticus, Deuteronomy–Joshua son of Nun, Judges, Ruth, the four Kingdoms, the two Chronicles, the Psalms of David, the Proverbs (or the Wisdom) of Solomon, Ecclesiastes, Song of Songs, Job, the Prophets: Isaiah, Jeremiah, the Twelve in one book, Daniel, Ezekiel, Esdras
The rabbis taught: “The order of the prophets is as follows: Jehoshua, Judges, Samuel, Kings, Jeremiah, Ezekiel, Isaiah, and the Twelve Prophets.” Let us see: Hosea, of the Twelve Prophets, was before Isaiah, as it is written [Hosea, i. 2]: “The beginning of the word of the Lord,” etc. This certainly cannot be understood that he was the first of the prophets to whom the Lord spoke since the time of Moses, as there were many prophets after Moses preceding Hosea. And therefore R. Johanan explains that he was the first of the four prophets who prophesied at that period; viz.: Hosea, Isaiah, Amos, and Micah. Hence he was before Isaiah. Why is he placed after? Because his book is counted among the Twelve, among whom were Haggai, Zechariah, and Malachi, who were the last of the prophets: therefore his book is placed together with theirs. But why was the book of Hosea not separated, and placed first? Because his book is small, and if it were placed separately it would become lost. However, was not Isaiah before Jeremiah and Ezekiel? Why is he not placed first? Because “Kings” ends with the destruction of the Temple, and the whole book of Jeremiah speaks of the destruction, and that of Ezekiel at the beginning speaks of the destruction and at the end of consolation, while Isaiah’s entire book speaks of consolation: destruction was put next to destruction, and consolation next to consolation.
The order of the Hagiographa is as follows: Ruth, Psalms, Job, Proverbs, Ecclesiastes, Song of Solomon, Lamentations, Daniel, Book of Esther, Book of Ezra, and Chronicles.
And who wrote all the books? Moses wrote his book and a portion of Bil’am [Numbers, xxii.], and Job. Jehoshua wrote his book and the last eight verses of the Pentateuch beginning: “And Moses, the servant of the Lord, died.” Samuel wrote his book, Judges, and Ruth. David wrote Psalms, with the assistance of ten elders, viz.: Adam the First, Malachi Zedek, Abraham, Moses, Hyman, Jeduthun, Asaph, and the three sons of Korach. Jeremiah wrote his book, Kings, and Lamentations. King Hezekiah and his company wrote Isaiah, Proverbs, Songs, and Ecclesiastes. The men of the great assembly wrote Ezekiel, the Twelve Prophets, Daniel, and the Book of Esther. Ezra wrote his book, and Chronicles–the order of all generations down to himself. [This may be a support to Rabh's theory, as to which, R. Jehudah said in his name, that Ezra had not ascended from Babylon to Palestine until he wrote his genealogy.] And who finished Ezra’s book? Nehemiah ben Chachalyah.