Thanks for the replies, people! Yes, it's true that the Bible says nothing about baptizing in the name of the Father and the Holy Spirit, but it also doesn't say how to baptize, the words one uses, and who can baptize. For example, can a woman baptize? How about in an emergency? How about blessing the emblems of the Sacrament? The Bible also says nothing about how to do that, as well, nor does it teach that the laying on of hands for the gift of the Holy Spirit was officially abolished.
The church must have paperwork and records, so that may be a reason it wants to have a record of the baptisms.
But the thing that got me thinking was whether people in the church can baptize their children? If you have a son or daughter who isn't baptized, but wants to be, I'd think his or her Father would be the one who would want to do it.
Has anyone here baptized a convert, a family member, etc.??