Sin came to the world through a woman, and just the same, salvation came through a woman. If Jesus is the New Adam, then Mary is the New Eve. Remember Genesis 3:15? What do you think the significance of that prophecy is?
Luke (through Elizabeth) calls her the mother of God. From there it’s a logical following: she’s the mother of the man who built the Church, therefore, she’s the mother of the Church.
Ignatius was the first person to imply Mary is the New Eve.
“There is one Physician who is possessed both of flesh and spirit; both made and not made; God existing in flesh; true life in death; both of Mary and of God; first possible and then impossible, even Jesus Christ our Lord." Ignatius, To the Ephesians, 7 (c. A.D. 110).
Justin Martyr was the first person to specifically label her as such.
“He became man by the Virgin, in order that the disobedience which proceeded from the serpent might receive its destruction in the same manner in which it derived its origin. For Eve, who was a virgin and undefiled, having conceived the word of the serpent, brought forth disobedience and death. But the Virgin Mary received faith and joy, when the angel Gabriel announced the good tidings to her that the Spirit of the Lord would come upon her, and the power of the Highest would overshadow her: wherefore also the Holy Thing begotten of her is the Son of God; and she replied, ‘Be it unto me according to thy word.’ And by her has He been born, to whom we have proved so many Scriptures refer, and by whom God destroys both the serpent and those angels and men who are like him; but works deliverance from death to those who repent of their wickedness and believe upon Him.” Justin Martyr, Dialogue with Trypho, 100 (A.D. 155).
I think you can derive all of this just from scripture, though. Jesus calls her the mother of John, why do you think John would mention that? There are thousands of details he could’ve put in his Gospel instead of that one line, yet he chooses to put in Jesus’ praise of Mary. That speaks to me.