Like many other Protestant Reformers, Martin Luther had no problem honoring Mary as the ‘Mother of God.’
Following the Ecumenical Council of Ephesus (A.D. 431), he was adamant that Mary be referred to and honored as the Θεοτοκος or ‘Birth-giver of God’ (also, ‘Mother of God’). The importance of this title is chiefly in its refutation of the heresy of Nestorianism, and it says perhaps more about the person of Jesus Christ than it does about Mary.
Christ was not divided between his human and divine natures, with Mary only mother of his humanity. The fathers at Ephesus make it clear that Jesus is one, undivided person, but without a mixture or dissolution of his divinity and humanity. He is fully God and fully man, and so Mary is truly the Mother of God—Mother of the God-man, Jesus Christ.
In his refutation of the Eucharistic views of both Caspar Schwenkfeld and Valentine Krantwald (Luther’s Works, vol. 37, p. 292), Luther associates Mary as the ‘Mother of God’ as a standard article of the Christian faith, alongside both Baptism and the sending of the Holy Spirit:
If everything that we are to believe were in the Old Testament, what need would there be of the New? Why should Christ have had to come to earth to teach us? In this way I could also say: Baptism is nothing; the sending of the Holy Spirit is nothing; that the mother of God is Mary is nothing; in short, no article of the Christian faith would stand.
In On the Councils and the Church, ca. 1539 (vol. 41, p. 104), Luther discusses the Nestorian controversy. In that work, he mentions the widely-accepted dogma of Mary as Theotokos:
Thus the histories relate that it was resolved in this council, in opposition to Nestorius, that Mary should be called Theotokos, “bearer of God,” even though Nestorius denied to God in Christ all idiomata of human nature such as dying, the cross, suffering, and everything that is incompatible with the Godhead. This is why they should not just have resolved that Mary was theotokos, but also that Pilate and the Jews were crucifiers and murderers of God, and the like.
In his collected works, Luther makes at least ninety more references to the ‘Mother of God.’ For example, in his sermons on John (vol. 22, pp. 23–24):
Everything else that a mother imparts to a child was imparted by Mary, the mother of God’s eternal Son. Even the milk He sucked had no other source than the breasts of this holy and pure mother.
And also, in A Sermon On Preparing to Die (vol. 42, p. 113), he writes:
[L]et no one presume to perform such things by his own power, but humbly ask God to create and preserve such faith in and such understanding of his holy sacraments in him. He must practice awe and humility in all this, lest he ascribe these works to himself instead of allowing God the glory. To this end he must call upon the holy angels, particularly his own angel, the Mother of God, and all the apostles and saints, especially since God has granted him exceptional zeal for this.