Seriously, is your mother my mother-in-law! :o My mother-in-law believes the same way and is Pentecostal. My husband is a convert. Here are some things we learned from talking to her and some scriptures to use.
When Peter says to baptized in the Name of Jesus Christ in Acts 2:38, it is not an exact code like in Mt 28:19 where Jesus himself says to baptized in the name of the father and of the son and of the holy spirit. If baptizing in the Name of Jesus Christ was the formula, why change it every once in awhile. There are two times in Acts where it says to baptized in the Name of the Lord Jesus (Acts 8:16, Acts 19:5). Like someone had said above, they had said it like this to differentiate which baptism they were doing since baptizing. Acts talks about John’s baptism a few times, one being in Acts 19:3.
Now if she is like my mother-in-law, she will say in reaction to Mt 28:19: “Well who is the father? Jesus. Who is the son? Jesus. Who is the holy spirit? Jesus. Jesus obviously meant to baptized in Jesus’s name.” This is interesting to me since it does not make sense with other scriptures. In Mt 3:18 it God says after Jesus is baptized by John, "
And a voice came from the heavens, saying, ‘This is my beloved Son, with whom I am well pleased.’" How could that have happened if there was only God is Jesus? Many times in the bible it states that Jesus will be or is seated at the right hand of God (such as Lk 22:69). How could that be if they are only one? Finally in Mark 13:32 it says, “But of that day or hour, no one knows, neither the angels in heaven, nor the Son, but only the Father. Be watchful! Be alert! You do not know when the time will come.” How does the Father know and not the son? There is also many instances that Jesus is praying to God like in John 17. They are three persons in one and it is a mystery on how it works. What sort of helped with his mom of recently is using an egg as an example. I don’t know if it is actually theologically sound, but we said it is like God being the yoke, Jesus being the egg white, and the holy spirit being the shell. Three distinct parts in one.
Well, that is all I have for now! Let me know if that helps.