According to both the Bible and history, the New Testament church invoked the name of Jesus at water baptism. Its baptismal formula was "in the name of Jesus Christ" or "Lord Jesus," not "in the name of the Father, Son, and Holy Ghost."
The Scriptural Record
Every time the Bible records the name or formula associated with an actual baptism in the New Testament church, it describes the name Jesus. All five such accounts occur in the Book of Acts, the history book of the early church. It records that the following people were baptized in Jesus' name.
The Jews, "Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost" (Acts 2:38).
The Samaritans. "They were baptized in the name of the Lord Jesus' (Acts 8:16).
The Gentiles. "And he commanded them to be baptized in the name of the Lord" (Acts 10:48). (The earliest Greek manuscripts that we have say, "In the name of Jesus Christ," as do most versions today.)
The disciples of John (rebaptized). "They were baptized in the name of the Lord Jesus" (Acts 19:5).
The Apostles Paul. "Arise, and be baptized, and wash away thy sins, calling on the name of the Lord" (Acts 22:16).
Moreover, the Epistles contain a number of references or allusions to baptism in Jesus' name. See Romans 6:3-4; I Corinthians 1:13; 6:11; Galatians 3:27 ; Colossians 2:12; James 2:7.
The only verse of Scripture that anyone could appeal to in support of a threefold baptismal formula is Matthew 28:19, in which Jesus commanded baptism "in the name of the Father, and of the Son, and of the Holy Ghost." The word name in this verse is singular, however, indicating that the phrase describes on supreme name by which the one God is revealed, not three names of three distinct persons.
The apostles understood Christ's words as a description of His own name, for they fulfilled His command by baptizing in the name of Jesus. There is only one God (Deuteronomy 6:4), and He has one supreme name today (Zechariah 14:9). Jesus is the incarnation of all the fullness of the Godhead (Colossians 2:9). Jesus is the name of the Son (Matthew 1:21), Jesus is the name by which the Father is revealed to us (John 5:43; 10:30; 14:9-11), and Jesus is the name in which the Holy Spirit comes (John 14:16-18, 26).
Luke 24:47 is a parallel verse to Matthew 28:19, and describes Jesus as saying that repentance and remission of sins-and baptism is for the remission of sins (Acts 2:38)-would be preached "in his name." Jesus is the only saving name, the name in which we receive remission of sins, the highest name made known to us, and the name which we are to say and do all things (Acts 4:12; 10:43; Philippians 2:9-11; Colossians 3:17).
Thus the one supreme, saving name of Matthew 28:19 is Jesus. We are to fulfill the command of that verse as the early church did, by invoking the name of Jesus at baptism.
The Historical Record
Respected historical sources verify that the early Christian church did not use a threefold baptismal formula but invoked the name of Jesus in baptism well into the second and third centuries.
Encyclopedia of Religion and Ethics (1951). II, 384, 389: "The formula used was "in the name of the Lord Jesus Christ" or some synonymous phrase; there is no evidence for the use of the triune name… The earliest form, represented in the Acts, was simple immersion… in water, the use of the name of the Lord, and the laying on of hands. To these were 
Interpreter's Dictionary of the Bible (1962), I 351: "The evidence… suggests that baptism in early Christianity was administered, not in the threefold name, but 'in the name of Jesus Christ' or 'in the name of the Lord Jesus.'"
Otto Heick, A History of Christian Thought (1965), I, 53: "At first baptism was administered in the name of Jesus, but gradually in the name of the Triune God: Father, Son, and Holy Spirit.
Hasting's Dictionary of the Bible (1898). I, 241: "[One explanation is that] the original form of words was "into the name of Jesus Christ" or 'the Lord Jesus,' Baptism into the name of the Trinity was a later development."
Williston Walker, A History of the Christian Church (1947), page 58: "The trinitarian baptismal formula,,, was displacing the older baptism in the name of Christ."
The New Schaff-Herzog Encyclopedia of Religious Knowledge (1957), I, 435: "The New Testament knows only baptism in the name of Jesus… which still occurs even in the second and third centuries."
Canney's Encyclopedia of Religions (1970), page 53: "Persons were baptized at first 'in the name of Jesus Christ' … or 'in the name of the Lord Jesus'… Afterwards, with the development of the doctrine of the Trinity, they were baptized 'in the name of the Father and of the Son and of the Holy Ghost.'"
Encyclopedia Biblica (1899), I, 473: "It is natural to conclude that baptism was administered in the earliest times 'in the name of Jesus Christ,' or in that 'of the Lord Jesus.' This view is confirmed by the fact that the earliest forms of the baptismal confession appear to have been single-not triple, as was the later creed."
Encyclopedia Britannica, 11th ed. (1920), II 365: "The trinitarian formula and triune immersion were not uniformly used from the beginning… Baptism into the name of the Lord [was] the normal formula of the New Testament. In the 3rd century baptism in the name of Christ was still so widespread that Pope Stephen, in opposition to Cyprian of Carthage, declared it to be valid."
Christians today should use the biblical baptismal formula as found in the New Testament. Everyone should be baptized by immersion in the name of the Lord Jesus Christ for the remission of sins.
