The letter itself deals with a relatively minor problem. The minor problem is this. There were stories that alleged that Mary was of the tribe of Levi. Moreover, there was some minor corroboration of these stories in that Elizabeth was Mary's cousin, yet Elizabeth is described this way:
Luke 1:5 There was in the days of Herod, the king of Judaea, a certain priest named Zacharias, of the course of Abia: and his wife was of the daughters of Aaron, and her name was Elisabeth.
In the letter, James of Edessa's response is essentially one of sola Scriptura. Rather than addressing the matter via tradition, James of Edessa begins from a very simple syllogism.
1. Jesus descended from David.
2. Jesus descended from Mary.
3. Therefore, Mary descended from David.
I would say he mistakenly analyzes the gospel genealogies in saying that neither of them is Mary's, but nevertheless, his fundamental approach of relying solely on Scripture to answer the question is correct.
He provides extensive documentation from the Old Testament to prove that the Messiah was the Son of David. From that, using the syllogism I identified above, he provides Mary's Davidic genealogy (with a lacuna from David to Mary) by simple logic.
Along the way, some of his comments about Scripture are quite fascinating - they are the sort of thing one would expect to hear from Calvin:
1. The divine apostle Paul when he wrote to men like those today, who were interpreting with great difficulty the mysterious words of the Gospel of Christ, said with wisdom and profundity: If there are men for whom our gospel is hidden, it is so for those who are perishing, for the unbelievers whose minds the god of this world has clouded, so that they do not see shining the light of the Gospel of the glory of the Messiah who is the image of God [2 Corinthians 4:3-4]. — But all of us, he said, who behold as in a mirror the glory of the Lord, are changed into the same shape, that is to say into the same image, from glory to glory, when we believe without hesitation in the words of the Gospel as (coming) from the Lord who is the Spirit [2 Corinthians 3:18]. So your Fraternity will know that, if someone interprets with great difficulty, finds obscure the words of the Gospel and does not believe it, or wants to oppose it with insidious and treacherous subtleties, this is one of those for whom the Gospel is hidden, one of those who are perishing, and one of those whom God has abandoned and, so that their minds are darkened by the shadows of the god of this world, and that the light of his Gospel does not shine in their eyes, because they are not worthy of being saved, because of their disbelief and the lack of rectitude of their spirit.- James of Edessa, Letter to John the Stylite, Sections 1-2
2. For we who — in the words of the apostle who knew and taught the mysteries of the Messiah — see and look at the words of the holy book as in a mirror, there is nothing in the Gospel that lends to difficulty or disbelief, but we are informed by the mysterious words it contains; we look at these words like one who looks at his reflection in a mirror, and they show clearly the picture of the truth. We learn from it that the Messiah has truly come, and we say that if he has truly come he was born, in the flesh, of the seed of David, as the prophets said of him; if he came and if he was born of the seed of David, he must necessarily also come at his time; if he came and he came at his time, and if he is born in the flesh of the seed of David, then the woman who gave birth is also by absolute necessity of the seed of David, as all these things depend on each other and are attached like (the rings) of a chain; they are combined and established by a necessary sequence of ideas and there is no (possible) hesitation on this subject.
The quotation provides a very typical sola Scriptura contrast between those who don't understand the Scriptures because their minds are darkened and those for whom the Scriptures are perspicuous - at least on certain subjects.
But James of Edessa goes on:
3. So by a firm and conclusive syllogism we must show to a Christian or Arab interlocutor that the Blessed Virgin Mary and Mother of God is of the seed of David, although this is not shown by the (holy) Books. This cannot be demonstrated by providing "evidence" from strange and superfluous stories that are brought forward by many and is written and read, but are not part of the holy books. Know indeed, O friend of truth, that I know the stories written by men of zeal, based on their own ideas without any testimony of the (holy) books. The Blessed Virgin Mary, mother of the Messiah, is the daughter of Anne and the just Joachim; the latter, according to the authors of these stories, was the son of Panthir, and Panthir was the brother of Melchi; the latter was the son of Iani, who was descended through his family from the tribe of Levi. They were based in Galilee, near the place where Tiberias was built.- James of Edessa, Letter to John the Stylite, Sections 3-4
As I said earlier, I do not want to demonstrate the matter with a superfluous demonstration taken from strange things, but using this firm and conclusive syllogism as I did earlier, O friend of God and the truth, that I want the truth to be upheld and not using words (learned) from superfluous stories; whoever the man may be who speaks with you and asks you or presses you on this issue, whether a Christian or a Mohammedan, if he is intelligent and has a reasonable soul, he will understand what the syllogism from which he will understand and will gladly testify to the truth without discussion. What I have said is enough to show clearly to a Christian or to a Mohammedan who talks about this subject, that the Blessed Virgin Mary is of the seed of David.
4. I must now quote for you the words of the prophets. They will show you evidence that the Messiah is, in the flesh, of the seed of David; it will then be shown you, in order to refute the Jews, that the Messiah has come in his time as it was written about him. Thus the expectation (of the Jews) is pointless, for because of their wickedness and blindness of heart, they were led to believe in a lie and not in the truth. The (holy) Spirit (by the) psalmist, testifies to you that the Messiah comes from David, when he says:
and speaking of the Messiah who is to come from him, he adds further:
I will make his seed eternal, and his throne like the days of heaven [Psalm 89:29 Peshitta].
The prophet Micah said:
And you Bethlehem, house of Ephrata, you're are not the (most) little among the thousands (the major cities) of Judah, because from you I will make come a ruler, who will command over Israel, and his outgoing (his origin is) from the beginning of the days of the world [Micah 5:2 (Unknown version)].
— Isaiah said:
— Jeremiah said:
Behold, the days come, says the Lord, and I will raise unto David a just offspring, and a King shall reign and be wise; he will make judgement and justice on the earth. In his days (during his reign) Judah will be saved, and Israel shall dwell with confidence and the name that the Lord will give it will be: our righteousness [Jeremiah 23:5-6].
— And after something else:
I will break the yoke on their necks and I will cut their chains, and foreigners shall not subjugate them any more, but they will serve the Lord their God and David their king, whom he has established [Jeremiah 30:8-9].
— And further on again:
Thus says the Lord: If you can make pointless my covenant that I have made with the day and my covenant that I have made with the night, so that day and night no longer have a place in time, then my covenant that I made with David my servant also will be pointless, and he shall have no son to reign on his throne [Jeremiah 33:20-21].
— Ezekiel says in the name of the Lord:
I will make a covenant together with David, I will kill every wicked animal on earth [Ezekiel 34:25 (LXX)].
By "animal" he means, in my opinion, the devil. — And further on:
I will purify them and they shall be my people, and I will be their God; and my servant David shall be their king and they shall have only one shepherd [Ezekiel 37:23-24].
— And again:
My servant (David) will be their leader forever [Ezekiel 37:25].
These are the testimonies of the prophets, to show you that the Messiah is descended from the seed of David, and there are others like them.
Notice How James of Edessa does not want to build any doctrine on those books. What you will find interesting, perhaps, is that it is from that class of books that Rome finds its earliest evidence for some of its doctrines about Mary. May I suggest that whatever you may think of James of Edessa's conclusions, his approach is better than theirs. He builds his theology on necessary inferences from Scripture, they build theirs without such evidence.
There is one final item from James of Edessa that I wish to bring to your attention:
So now, everyone: the true Christians and the bold heretics, the Muslims and the Jews in spite of themselves, all confess truly and necessarily that the Messiah has indeed come, that he came in his time, and he is descended from the seed of David, and if all these things are necessarily satisfied (are linked), Mary, the Blessed Virgin who gave birth to him, is also descended from David, although that is not explicitly written in the Holy Book, and that we are not able to produce what is not written. Because that which the truth (the reasoning) states, without allowing anything to be added or subtracted, shows much better the truth to our spirit and to our faith than if we were gathering superfluous words which are not written (in the Bible) and that we can not demonstrate from the sacred books.- James of Edessa, Letter to John the Stylite, Section 6
That last portion really shows the force of James of Edessa's comment. His operative principle is sola Scriptura. If he can't show it from the Bible, he can't show it. He treats extra-scriptural tradition as unnecessary and extraneous words, and basically denigrates them to the point of their not being worth considering (although he knows what they say).
It would have been nice for James of Edessa to explicitly state the principles of sola Scriptura, but I believe we can fairly clearly see them at work in this letter. The entire letter may found at the following link (link).