In The Name Of (Eis To Onoma)

While there is nothing explicit in Matthew's Great Commission which adds decisively to our understanding of the essential meaning of baptism we have discovered that   baptism is inseparably bound up with costly  discipleship. It is for those who are committed, without reserve, to the lordship of Christ. We have also considered the commission as it was understood and implemented by the Twelve in the Book of Acts, i.e. the pattern in Acts anticipated by this commission: the proclamation of the Gospel, the response of faith, baptism, forgiveness, the gift of the Spirit, and devotion to the apostles' teaching. Baptism is inseparable from these ingredients. Moreover it is clear that the making and baptising of disciplesis unaffected by distinctions of race or nationality, and that the inclusion of 'all nations' is the fulfillment God's promise to Abraham and the outworkingof his covenant purposes.  

We can now go one step further and consider the significance of baptism "in the name of". The first thing for consideration is the use of the phrase in Rabbinic literature. Beasley-Murray gives three illustrations taken from Strack-Billerbeck. Baptism in the New Testament, Paternoster 1972, pp 90f; Strack-Billerbeck, Kommentar zum Neuen Testamentaus Talmud  und Midrasch, vol. 1, 1922, pp 1054f. 

(1) Heathen slaves on entry into a Jewish house were compelled to receive a baptism "in the name of slavery", i.e. to become slaves. Slaves being set free were to be immersed "in the name of freedom", i.e. to become free. On this analogy baptism in the name of God "sets the baptised in a definite relation to God; Father Son and Holy Spirit become to the baptised what their name signifies".
(2) An offering is slaughtered in the name of six things, i.e. with respect to its intention, e.g. for the benefit of the offerer, for the sake of God, with regard to the altar fires, in view of the sweet savour. Hence a person is baptised "for the sake of God, to make the baptised over to God". Hence 'in the name of' equals 'with respect to, for the benefit of, for the sake of'.
(3) A Samaritan must not circumcise an Israelite because the Samaritans circumcise" in the name of Mount Gerizim", with the obligation of venerating the God of the Samaritans who is worshipped there.

It is worth extrapolating from this Rabbinic understanding that circumcision in the name of 'Mount Gerizim' points to 'obligation' and that an offering slaughtered in the name of signifies 'for the sake of'. According to these parallels baptism in the name of the Trinity signifies 'obligation to' and 'for the sake of' the Trinity in whose name the disciple is baptised. Perhaps, however, the most significant lesson is to be learned from the freeing of a slave through circumcision in the name of freedom. By this parallel the Trinity becomes to the person baptised all that is signified by the name of the Trinity. Baptism then represents the deepest, most intimate and most profound relationship between the disciple and God. We shall return to this shortly under the heading 'the Father, the on and the Holy Spirit'.

The second thing for consideration has to do with the use of the preposition 'in'. This translates the Greek word eis which basically means "into". Luke in his Book of Acts uses this same phrase ("in the name") four times in connection with baptism, but uses three different prepositions (eis, into; en, in; epi, upon) all of which are translated "in". There is another important difference. Hereas Luke seems to use his three prepositions synonymously, Matthew seems never to confuse them. Prepositions and Theology inthe Greek New Testament, Paternoster's Dictionary of Dictionary of New Testament TheologyNew Testament Carson comments: 

"Those who become disciples are to be baptised eis ('into', NIV margin) the name of the Trinity. Matthew, unlike some NT writers, apparently avoids the confusion of eis (strictly 'into') and en (strictly 'in'…) common in Hellenistic Greek ; and if so, the preposition 'into' strongly suggests a coming-into-relationship-with or a coming-under-the Lordship-of (cf. Allen; Albright and Mann)… It is a sign both of entrance nto Messiah's covenant community and of pledged submission to his lordship.

This understanding of eis confirms, therefore, a parallel with ideas present in Rabbinical literature, especially the idea of a deep and profound relationship into which a person enters when they becomes a disciple.