According to Moulton & Milligan,
The phrase εἰς (τὸ) ὄνομά τινος is frequent in the papyri with reference to payments made “to the account of any one”.... The usage is of interest in connexion with Mt 28:19, where the meaning would seem to be “baptized into the possession of the Father, etc.”1
This is corroborated by BDAG, which also notes that this could be a Semitic idiom for לְשֵׁם:
[With] εἰς: somet[hing] evidently as rendering of rabb[inical] לְשֵׁם with regard to, in thinking of.... for his sake, or we have here the frequently attested formula of Hellenistic legal and commercial language.... εἰς (τὸ) ὄν. τινος to the name=to the account (over which the name stands). Then the deeds of love, although shown to humans, are dedicated to God.—The concept of dedication is also highly significant, in all probability, for the understanding of the expr[ession] βαπτίζειν εἰς (τὸ) ὄν[ομα] τινος. Through baptism εἰς (τὸ) ὄν[ομα] τ[ινος] those who are baptized become the possession of and come under the dedicated protection of the one whose name they bear.2
These lexical resources cite a 1903 work by Wilhelm Heitmüller entitled Im Namen Jesu. His position (and that of a number of scholars) has been summarized as follows:
[T]he person baptized was dedicated to Jesus, having become his property.... [In the papyri, Heitmüller] found 'into the name' used in Graeco-Hellenistic banking terminology, the 'name' being that of a person to whose account something was credited. These observations naturally led to the interpretation that Jesus is the heavenly Kurios to whose ownership the baptized person was transferred.3
Heitmüller (among others) has been criticized for assuming that this banking context should be inferred every time this construction is used.4 Lars Hartman contended for the first sense represented by BDAG above (i.e., that it is a Semitic idiom translating לְשֵׁם), noting it would have both "positive content as well as a negatively demarcating meaning," the former uniquely identifying the rite of Christian baptism and the latter differentiating it from John's baptism. He also noted that it could also have been understood differently in various regions.5
Given this information, I believe that in the context of Matthew 28:19, εἰς τὸ ὄνομα likely carried some or all of these connotations for early Christian readers (depending on whether they primarily spoke Greek or Hebrew/Aramaic). The phrase εἰς τὸ ὄνομα
- uniquely identifies this as the rite of Christian baptism when invoked with the Trinitarian formula,
- demarcates it from John's baptism,
- refers to being baptized into the possession of the Father, Son, and Holy Spirit; and
- implies coming under God's dedicated protection.
1 James Hope Moulton and George Milligan, The Vocabulary of the Greek Testament (London: Hodder and Stoughton, 1930), 451. Read online.
2 William Arndt et al., A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), 713.
3 Lars Hartman, "Into the name of Jesus: a suggestion concerning the earliest meaning of the phrase," New Testament Studies 20, no. 4 (July 1974): 432. https://doi.org/10.1017/S002868850001225X
4 Ibid., 433.
5 Ibid., 435, 440.