ANSWER: First, it should be noted that the Bible uses several different names and titles for the Lord. Your question concerns, however, whether the phrase "in the name of Jesus" should be used during a person's baptism or whether other phrases that include the Father, or even the Holy Spirit, should be utilized.
Your question concerning the exclusive use of the name (authority) of Jesus while baptizing people is a good one. It has troubled quite a few people. The Bible gives us clear examples of how baptism was conducted in the early church.
And he commanded them to be baptized in the name of the Lord. Then they besought him to remain for a number of days. (Acts 10:48, HBFV throughout unless noted).
And after hearing this, they were baptized into the name of the Lord Jesus. (Acts 19:5, see also Acts 8:15 - 16, Acts 2:38).
Many passages in the New Testament discuss "the name of the Lord" or Jesus being used as the authority to preach the gospel, or to ask for healing or to call upon him in order to be saved (Acts 2:21, 9:29, Romans 10:13, James 5:14).
Scripture does not endorse or record any example of "the name" or authority of the Holy Spirit being used to perform a baptism. This makes sense when one understands the Bible teaching that God's Spirit is not a separate or distinct person in the Godhead (see our article on the Trinity).
The baptism of a person in order to ask God to give him or her his spirit should be done "in the name of Jesus." When the Apostle Peter was questioning by Jewish religious leaders regarding by what authority a man was healed and the gospel preached he stated, "Be it known to you all, and to all the people of Israel, that in the name of Jesus Christ the Nazarean . . . by Him this man stands before you whole" (Acts 4:10).
In closing, the Apostle Paul teaches us by what authority or power we should do all things as a Christian. He told the church in Colossae, "And in everything - whatever you do in word or in deed - do all in the name of the Lord Jesus" (Colossians 3:17). We can never go wrong by following this sacred admonition.