Exodus 20:7 says, "Thou shalt not take the name of the LORD thy God in vain; for the LORD will not hold him guiltless that taketh his name in vain." This is one of the ten commandments, so it is very important as part of the moral code of the Bible for all humanity. It is part the Judeo-Christian ethic, and did not get fulfilled or superseded by anything in the New Testament. However, most people misunderstand the spiritual implications it has for us today. Most people think this means that when you hit your thumb with a hammer, you should not use the name of the Lord as a curse word. This is certainly true. In fact, a man was stoned to death for doing this (Lev 24). Yet the meaning goes far deeper. The word "take" in this verse is the Hebrew word "nasa", which means "to bear continually, to carry, to lift up, and to exalt". Vanity in the Bible does not mean being overly concerned with one's appearance. It means emptiness or lack of substance. The book of Ecclesiastes is a 12 chapter illustration of what the Bible means by vanity. Therefore what Exodus 20:7 is really saying to us is, "Do not continually bear the name of the Lord superficially without meaning and substance."
There are two essential questions to ask about water baptism in the New Testament. Why are people baptized, and how are people baptized? There is only one reason given in the Bible to be baptized - for the remission of sins (Mk 1:4, Lk 3:3, Acts 2:38, Acts 10:43-48). There is also only one way to be baptized - by immersion in water in the name of Jesus Christ (Mt 3:16, Jn 3:23, Acts 2:38, Acts 8:36-39, Acts 10:43-48, Acts 19:1-7, Rom 6:1-4, Col 2:8-12). When we are baptized in the name of Jesus Christ, our sins are remitted (pardoned, forgiven, released), but there is more. We "take" the name of Jesus Christ much like a bride takes the name of her husband. The church is the bride of Christ (Jn 3:29, 2Cor 11:2, Eph 5:22-33, Rev 19:7, Rev 21:2 & 9, Rev 22:17). As a bride identifies with her husband, submits to his leadership and authority, and becomes one with him (Gen 2:24, Mt 19:1-7, Eph 5:22, Col 3:18), we become the bride of Christ. In the days of the Bible, a woman was "espoused" to her husband before the marriage was consummated and finalized. We compare this to a modern engagement, but the espousal was legally binding and required a divorce to be broken (Mt 1:18-19). We are joined to Christ, although the marriage has not been finalized.
When James rendered his decision in the debate about Gentiles being circumcised, he quoted Amos 9:11-12 which speaks of the "heathen which are called by my name". James quoted it as "the Gentiles upon whom my name is called". This occurs at water baptism (Acts 22:16). We are given spiritual authority to act with power of attorney on behalf of the kingdom of God through the name of Jesus Christ (Mt 18:20, Mk 9:37-41, Mk 16:17, Jn 14:13-14, Jn 15:16, Jn 16:23-26, 2Cor 5:17-20).
Children take the name of their father. The name of God the Father is not Father. It is Jesus Christ (Jn 5:43). That is because Jesus Christ is God the Father manifest in the flesh (Jn 1:1-3 & 14, Jn 14:7-11, Jn 10:30, 2Cor 5:19, Col 2:9, 1Tim 3:16). Christians are the children of God (Jn 1:12, Rom 8:14, 2Cor 6:16-18, Phil 2:15, 1Jn 3:1-2). Children have special status with their father that others do not enjoy (Mt 7:11, Mt 17:25-26, Mk 7:27, Lk 11:17, Lk 20:36, Rom 8:16-21, Eph 1:5). The name of Jesus Christ is the family name of the people of God (Eph 3:15).
We are given great access to what heaven offers through the name of Jesus Christ. This also means we are given great responsibility to access it and use properly according to the will and word of God. People should take being baptized in the name of Jesus Christ seriously, understanding that God expects them to be committed to follow through with their relationship with Him forever. Anything less is taking His name in vain. If we call ourselves by the name of Christ, our life should reflect that.