What criteria do Catholics use to determine if an old testament character prefigures Mary?
In other words, what you are asking is what the Church uses as a characterizing mark or trait between of Mary and Old Testament characters that prefigure the Mother of Jesus, such as Old Testament Types of Mary.
Thus it is obvious that certain biblical women prefigured Mary simply because they are female biblical personages. In another sense some other biblical object could prefigure Our Lady due to some physical traits that can be viewed in common between Mary and some other biblical object such as the Ark of the Covenant.
Here is the list of fourteen feminine Old Testament figures, beginning with Eve and ending with Bathsheba. Their portrait, will be followed by a comparison between each of the Old Testament figures and Mary: Eve, Sarah, Rebekah, Rachel, Leah, Deborah, Jachobed, Miriam, Judith, Esther, Tamar, Rahab, Ruth and Bathsheba:
Eve: Mother of all the living.
Sarah: Abraham's wife, who brings the promise to fruition and helps Abraham to live out his faith in God.
Rebekah: The second matriarch of Israel. "The girl was very beautiful, a virgin, untouched by man."
Rachel: She is the most genuine of the Israelite tribes issue, Hence she is a woman of Israel par excellence since she is the mother of Joseph and Benjamin by Jacob. "The story of Rachel is a story of unparalleled love and devotion in the biblical narrative." [See: Anchor Bible, vol 5, 605.]
Leah: She is an important mother of the Israelites. She stems from Terah of Mesopotamia through Nahor and Bethuel. Her father is Laban, son of Bethuel and brother to Rebekah. Leah is the mother of Reuben, Simeon, Levi, Judah, Issachar, Zebulus, and Dinah.
Deborah: Judges 5 is the important chapter for remembering her, while Pseudo-Philo offers fascinating information about her in the Tradition of Israel.
Jochebed: The mother of Moses, Miriam, and Aaron, is considered as a "Mother of Israel" in the Jewish tradition. She is a Levite and is mentioned in the genealogies of Exodus 6:20 as the wife of Amram and as the mother of Moses and Aaron. In Numbers 26:59 she is described as "of the tribe of Levi, born to the tribe in Egypt."
Miriam: Undoubtedly, the name of the Blessed Virgin Mary was given to her by her parents honoring the great person of Miriam, the sister of Moses and Aaron (in the Old Testament).
Judith: She is the heroine of the deutero-canonical book with the same name. She exemplifies the ideal woman of later Jewish piety (150-100 BCE). In many of the events of her life she fits the description of a woman who was a Pharisee.
Esther: She is the heroine and is the paradigm for a fully liberated woman who places all her confidence in God. Through prayer and fasting she is able to challenge the evil perpetrated by the Persians and to intercede for her people Israel before King Ahasuirus.
Tamar: Her name in Hebrew means "Palm Tree". She is the first woman named in the genealogy of Matthew: "Judah became the father of Perez and Zerah, whose mother was Tamar" (Mt 1:3). The source for the mention of Tamar is Genesis 38.
Rahab: There are several points of convergence between the stories of Rahab and Mary; these help us gain a greater understanding of both women, and the faith that has linked them in the biblical tradition: sexuality is an issue in both stories; both women ran the risk of punishment (death); both were the means through which God took possession of the land and of human hearts; both were signs and exemplars of faith (Heb 11); both were mother to the household of faith.
Ruth: The Church, too, in its earlier tradition continues the typology showing Boaz as a figure of Christ while Ruth images the Church. In the twelfth and thirteenth centuries Mary also is seen to be symbolized in Ruth. Peter of Celle (1115-1182) sees a parallel in Ruth's words, "I am Ruth, your handmaid," with Mary's, "Behold the handmaid of the Lord, be it done to me according to your word" (Lk 1:38).
Bathsheba: In the Jewish tradition contemporaneous with the formation of the New Testament, Bathsheba is seen as a noble woman of Israel. As Queen Mother, Gebirah, she had great influence both with David and his successor Solomon.
Old Testament Types of Mary
In the Litany of Loreto the Church honours Mary as the Ark of the Convenant.
The Church fathers loved to delve into Holy Scripture and find deep meanings and parallels; or types and shadows. One of the titles they gave to the Blessed Virgin Mary was New Eve or Second Eve: because the first Eve said “no” to God and brought about original sin and the fall of mankind.
Mary the “second” Eve (being immaculate and without sin from the time of her conception and thus analogous to Eve as regards sinlessness), said “yes” to God at the Annunciation and in so doing, played a key role in bringing about our redemption, as the Mother of God the Son, our Lord Jesus Christ.
Another fascinating analogy along these lines is the notion of Mary as the ark of the new covenant. The original ark was a marvelous gold-lined wooden box [House of gold] that was the holiest item in Judaism. It contained the tablets of the Ten Commandments, and God made His presence especially manifest over the ark, above what was called the “mercy seat” (Ex 25:22): between two carved golden cherubim (angels). This was an early parallel to the eucharistic Real Presence.
The Church Fathers drew from the following biblical passages in developing this belief:
Luke 1:35 (RSV) And the angel said to her, “The Holy Spirit will come upon you, and the power of the Most High will overshadow you; therefore the child to be born will be called holy, the Son of God.”
The Greek word for overshadow is episkiasei, which describes a bright, glorious cloud. It is used with reference to the cloud of transfiguration of Jesus (Mt 17:5; Mk 9:7; Lk 9:34) and also has a connection to the shekinah glory of God (Ex 24:15-16; 40:34-38; 1 Ki 8:10).
Mary is, therefore, in effect, the new temple and holy of holies, where God was present in a special fashion. Scripture draws many parallels between Mary, the “ark of the new covenant” and the ark of the (old) covenant:
Exodus 40:34-35 Then the cloud covered the tent of meeting, and the glory of the LORD filled the tabernacle. And Moses was not able to enter the tent of meeting, because the cloud abode upon it, and the glory of the LORD filled the tabernacle.
Amazing Parallels Between Mary and the Ark of the Covenant
Another possible prefigurement of Mary in the Litaniae lauretanae is that she is honour as the Turris Davidica or the Tower of David, although not self evident at first.
Tower of David is another title of the Virgin Mary that is not self-evident. Blessed John Henry Newman explains its meaning to us. King David is a type of Our Lord, and thus the Tower of David is a defence of Our Lord. And Mary is just that defence because all the veneration that she receives reinforces truths about his divinity. She does not eclipse Him; “He is infinitely above His Mother; and all that grace which filled her is but the overflowings and superfluities of His incomprehensible Sanctity.”
Newman argues that countries which pay veneration to the Mother of Jesus keep faith in her Son’s Divinity and Humanity.
Holy Mary, Tower of David, defend the Church, and pray for us that we may courageously teach and hold on to the truths about your Son.
Mary is the “Turris Davidica,” the Tower of David
A TOWER in its simplest idea is a fabric for defence against enemies. David, King of Israel, built for this purpose a notable tower; and as he is a figure or type of our Lord, so is his tower a figure denoting our Lord’s Virgin Mother.
She is called the Tower of David because she had so signally fulfilled the office of defending her Divine Son from the assaults of His foes. It is customary with those who are not Catholics to fancy that the honours we pay to her interfere with the supreme worship which we pay to Him; that in Catholic teaching she eclipses Him. But this is the very reverse of the truth.
For if Mary’s glory is so very great, how cannot His be greater still who is the Lord and God of Mary? He is infinitely above His Mother; and all that grace which filled her is but the overflowings and superfluities of His incomprehensible Sanctity. And history teaches us the same lesson. Look at the Protestant countries which threw off all devotion to her three centuries ago, under the notion that to put her from their thoughts would be exalting the praises of her Son. Has that consequence really followed from their profane conduct towards her? Just the reverse—the countries, Germany, Switzerland, England, which so acted, have in great measure ceased to worship Him, and have given up their belief in His Divinity while the Catholic Church, wherever she is to be found, adores Christ as true God and true Man, as firmly as ever she did; and strange indeed would it be, if it ever happened otherwise. Thus Mary is the “Tower of David.”