Why would she so easily discount the words of Jesus? What does she say as to the reason Jesus said this?
If Jesus said this how is it a change for the Catholic Church to still do this?
Perhaps the historical writings of early Christians (pre Constantine) would help your mom. These are from the earliest Christians, the persecuted underground church.
After the foregoing instructions, baptize in the name of the Father, and of the Son, and of the Holy Spirit, in living [running] water. If you have no living water, then baptize in other water, and if you are not able in cold, then in warm. If you have neither, pour water three times on the head, in the name of the Father, and of the Son, and of the Holy Spirit. Before baptism, let the one baptizing and the one to be baptized fast, as also any others who are able. Command the one who is to be baptized to fast beforehand for one or two days (Didache 7:1 [ca. A.D. 70]).
As many as are persuaded and believe that what we [Christians] teach and say is true, and undertake to be able to live accordingly, and instructed to pray and to entreat God with fasting, for the remission of their sins that are past, we pray and fast with them. Then they are brought by us where there is water and are regenerated in the same manner in which we were ourselves regenerated. For, in the name of God, the Father… and of our Savior Jesus Christ, and of the Holy Spirit [Matt. 28:19], they then receive the washing with water. For Christ also said, “Unless you are born again, you shall not enter into the kingdom of heaven” (First Apology 61 [A.D. 151]).
He [Jesus] came to save all through himself – all, I say, who through him are reborn in God; infants, and children, and youths, and old men. Therefore he passed through every age, becoming an infant for infants, sanctifying infants; a child for children, sanctifying those who are of that age . . . [so that] he might be the perfect teacher in all things, perfect not only in respect to the setting forth of truth, perfect also in respect to relative age (Against Heresies 2:22:4 [A.D. 189]).
[N]o one can attain salvation without baptism, especially in view of the declaration of the Lord, who says, “Unless a man shall be born of water, he shall not have life” (On Baptism 12:1 [A.D. 203]).
When we are about to enter the water — no, just a little before — In the church and under the hand of the bishop, we solemnly profess that we renounce the devil and his pomps and his angels. Thereupon we are immersed three times (The Crown 3:2 [A.D. 211]).
Where there is no scarcity of water the stream shall flow through the baptismal font or pour into it from above; but if water is scarce, whether on a constant condition or on occasion, then use whatever water is available. Let them remove their clothing. Baptize first the children, and if they can speak for themselves let them do so. Otherwise, let their parents or other relatives speak for them (The Apostolic Tradition 21:16 [A.D.215]).
Recognitions of Clement
But you will perhaps say, ‘What does the baptism of water contribute toward the worship of God?’ In the first place, because that which has pleased God is fulfilled. In the second place, because when you are regenerated and born again of water and of God, the frailty of your former birth, which you have through men, is cut off, and so . . . you shall be able to attain salvation; but otherwise it is impossible. For thus has the true prophet [Jesus] testified to us with an oath: “Verily, I say to you, that unless a man is born again of water . . . he shall not enter into the kingdom of heaven” (Recognitions of Clement 6:9 [A.D. 221]).