Where does the Bible authenticate re-baptizing anyone who believed in Jesus and was baptized in His name?
I know of none. Baptism is an act that is done once. "For you are all sons of God through faith in Christ Jesus. For as many of you as were baptized into Christ have put on Christ" (Galatians 3:26-27). Even when a person falls from grace, he is still considered a brother -- a fallen brother, but a brother none the less. "And if anyone does not obey our word in this epistle, note that person and do not keep company with him, that he may be ashamed. Yet do not count him as an enemy, but admonish him as a brother" (II Thessalonians 3:14-15).
However, there are many people who feel they were properly baptized, but in fact were not. For example, Paul ran across men in Ephesus who were baptized. However, they did not know about the Holy Spirit. Such could not be possible for those properly baptized since baptism is done in the name of the Spirit (Matthew 28:19) and a consequence of proper baptism is receiving the gift of the Spirit (Acts 2:38). The result was that these men who had been baptized into John's baptism were baptized again into the name of Jesus (Acts 19:1-7).
When one is baptized into the name of Jesus, it is not a formula that is recited that makes the baptism authentic. Acting in the name of Jesus indicates a person is not acting on his own authority but by the authority of Jesus. "And whatever you do in word or deed, do all in the name of the Lord Jesus, giving thanks to God the Father through Him" (Colossians 3:17). When Peter was asked by what authority he did what he did, Peter answered "by the name of Jesus Christ of Nazareth" (Acts 4:7-12). See the lesson "All in a Name" for more details. If being baptized into the name of Jesus simply meant invoking the name of Jesus, then the accusation that baptism is just a ritual that does not effect a believer's salvation would have some merit.
Herein then lies the foundation of the problem. Can a baptism performed for the wrong reasons be a proper baptism? Peter stated, "Corresponding to that, baptism now saves you--not the removal of dirt from the flesh, but an appeal to God for a good conscience--through the resurrection of Jesus Christ" (I Peter 3:21). Baptism is not a mere washing in water. Baptism saves because the person appeals to God for cleansing. It is the voluntary submission to God's command that allows the person to take advantage of the cleansing blood of Jesus. Can that cleansing take place if the person being baptized does not believe baptism saves?
There are many people in this world who received a ritual called baptism when they were infants. These infant baptisms are done in the name of the Father, Son, and Holy Spirit; but do these baptisms save the infant? Since the Bible gives no authority for the baptizing of infants, then we must conclude that such baptism cannot save because they are performed outside the authority of God. They are acts of lawlessness. Too, saving baptisms are combined with faith (Mark 16:16) and repentance (Acts 2:38). Since an infant is too young to express faith and too young to have committed sin from which he needs to repent, infant baptism does not bring salvation. People who were baptized as infants need to act within the law of God and be baptized again in order to be saved.
Several denominations practice a baptism of adults, but these adults are told that the baptism does not effect their salvation. Instead a slogan is given that baptism is the outward sign of an inward grace (a statement that is not found anywhere in the Bible). If you ask why these people were baptized, you are told that it is to fulfill the command of God and that it makes the one baptized a member of that particular denomination. The baptism taught in the New Testament was for the forgiveness of sins (Acts 2:38), it was to wash away sins (Acts 22:16), it was to free men from sin (Romans 6:3-7), and it was done to be saved (Mark 16:16; I Peter 3:21). Hence, these denominational baptisms are not the same as the baptism taught in the New Testament. Even though God's name is used during the procedure, the type of baptism practiced by these denominations is without the authority of God. Also, the baptism in the New Testament added people to Christ's church (Acts 2:41, 47; Galatians 3:27), it did not make them a part of a denomination because the concept of a denomination was forbidden (I Corinthians 1:10-15). Once again, these baptism are different from the baptism taught in the Bible.
There is only one true baptism (Ephesians 4:5). If a person received the baptism taught in the Bible, then there would be no need to be baptized again. However, many people have only received a pseudo-baptism. When they learn the truth, their faith in God's teachings will spur them to correct any deficiencies in their obedience to God's will in their past. The past does not matter. Faith in God declares that only the future is important. Since a pseudo-baptism is not authorized by God, they have not truly been baptized in the name of Jesus. Such people then need to be, not re-baptized, but truly baptized in the name of the Father, and of the Son, and of the Holy Spirit.