How would Catholics respond to the theory that Mary was a surrogate mother that had no DNA in common with Jesus?

The Catholic Church certainly teaches that Christ is the actual physical offspring of Mary; that is, that he is genetically descended from Mary:

The Holy Spirit, “the Lord, the giver of Life,” is sent to sanctify the womb of the Virgin Mary and divinely fecundate it, causing her to conceive the eternal Son of the Father in a humanity drawn from her own.

(Catechism of the Catholic Church, paragraph 485; emphasis added)

In fact, the One whom she conceived as man by the Holy Spirit, who truly became her Son according to the flesh [that is, her actual, physical son], was none other than the Father’s eternal Son, the second person of the Holy Trinity. Hence the Church confesses that Mary is truly "Mother of God" (Theotokos).

(Catechism of the Catholic Church, paragraph 495)

It appears from these two quotations, at the minimum, that the Church currently teaches that Mary was the mother of Jesus in the usual sense (that is, in being an actual "blood relative" of Jesus, genetically related to him).

In addition, Thomas Aquinas addresses this question in his Summa Theologica, though modern-day understanding of his discussion may be hampered by his incorrect understanding (derived from Aristotle) of prenatal development.

Aquinas, with Aristotle, believed that the "humanity" of a human before birth derived solely from the father, and that the mother's only physical contribution was providing the physical substance (the actual flesh, bones, and blood) of the child. In considering the question "Whether the flesh of Christ was conceived of the Virgin's purest blood?", Aquinas comments:

It belongs to the supernatural mode of Christ's generation, that the active principle of generation was the supernatural power of God: but it belongs to the natural mode of His generation, that the matter from which His body was conceived is similar to the matter which other women supply for the conception of their offspring.

(Summa Theologica, Third Part, Question 31, Article 5)

In other words, Aquinas believes that the actual physical substance of Jesus' body was in fact derived from Mary's body (even though he discusses it in terms we now know to be factually incorrect), so that he is her son (physically) as well as the Son of God (eternally).

Aquinas addresses the question even more directly shortly afterward. Question 35, Article 3, of the Third Part of the Summa asks "Whether the Blessed Virgin can be called Christ's Mother in respect of His temporal nativity?"—that is, whether Mary is in fact the physical mother of "God born into the world", in the sense of being his genetic ancestor, not just the vessel in which he was "incubated" until birth.

Aquinas, though he doesn't consider "surrogate motherhood" in the sense that we can think of it nowadays, does bring up the potential objection that miraculous births in general (like the "birth" or creation of Eve from the body of Adam) don't necessarily result in parent-child relationships. He answers the objection (quoting Saint John of Damascus) by saying,

As Damascene says ... "The temporal nativity by which Christ was born for our salvation is, in a way, natural, since a Man was born of a woman, and after the due lapse of time from His conception: but it is also supernatural, because He was begotten, not of seed, but of the Holy Ghost and the Blessed Virgin, above the law of conception." Thus, then, on the part of the mother, this nativity was natural, but on the part of the operation of the Holy Ghost it was supernatural. Therefore the Blessed Virgin is the true and natural Mother of Christ.

That is, it is the fertilization of Mary by the Holy Spirit that is supernatural or miraculous about Jesus' birth, but everything else about Jesus' conception and birth (including, as we'd now phrase it, his conception from an egg of Mary's) was totally natural—so that she was his mother in the usual sense.