The same phrase can also be found in Elements of Religious Philosophy, Preliminary to the Aphorism on Spiritual Religion (reprinted in The Collected Works of Samuel Taylor Coleridge, Volume 9: Aids to Reflection, Princeton University Press, 2017):
Omnia exeunt in mysterium, says a Schoolman: i.e. There is nothing, the absolute ground of which is not a Mystery. The contrary were indeed a contradiction in terms: for how can that, which is to explain all things, be susceptible of an explanation? It would be to suppose the same thing first and second at the same time.
A footnote to “... says a Schoolman” explains:
“All things go out into mystery”. The Latin phrase appears also in The Friend (CC) II 279, C continuing his id est, as here. A source is not traced.
The preceding footnote pointed out that Coleridge reproduces a long passage from The Friend. In The Collected Works of Samuel Taylor Coleridge, Volume 4 (Part II): The Friend also says in a footnote that the source for “Omnia exeunt in mysterium” is untraced.
The phrase is also used in many other sources that may have borrowed it from Coleridge. Below are a few examples.
Well does Emerson say, that the wise men are distinguisted from fools by their wondering at what is usual rather than at what is unusual. The old scholastics used to say, omnia exeunt in mysterium.
From The Penn Monthly Volume 5 (1874).
In a footnote to the phrase “all things run into the inscrutable”:
“Omnia exeunt in mysterium.” An equivalent thought is that of Dr. Tyndall (Belfast Address): “All we see around us, and well we feel within us, have their unsearchable roots in a cosmical life of which only an infinitesimal span is offered to the investigation of man.” (…)
Quoted from Behind the Veil: An Outline of the Bible Metaphysics Compared with Ancient and Modern Thought by Thomas Griffith (1876).
The phrase was even used in the context of discussions on science:
In this connection we may note that the question: “What is electricity?” has again been declared as an open one. Formerly it was regarded as a subtle fluid, distributed over all mundane surfaces, with some quality. Then it was said to be a force like heat. Now speculation has reached a kind of agnosticism.—Omnia exeunt in mysterium, the scholastics said.
Quoted from The American: A National Journal, Volumes 19-20 (1889).
Perhaps this confession will not wound you so sharply as it would have done twenty years ago; for I think you cannot have failed to notice that for some time hypotheses have been advanced by men of pure science which are nothing less than transcendental, and I suspect that most modern chemists and biologists of repute would not hesitate to subscribe the dictum of the old Schoolman, Omnia exeunt in mysterium, which means, I take it, that every branch of human knowledge if traced up to its source and final principles vanishes into mystery.
From "Novel of the White Powder" (1895) by Arthur Machen.
I take it, that the truth is that all human knowledge is subject to the same disadvantage, the same doubts and reservations. Omnia exeunt in mysterium was an old scholastic maxim; and the only people who have always a plain answer for a plain question are the pseudo-scientists, the people who think that one can solve the enigma of the universe with a box of chemicals.
From Hieroglyphics: A Note upon Ecstasy in Literature by Arthur Machen. Apparently, the phrase “Omnia exeunt in mysterium” is so closely associated with Machen that Brian J Showers could write in The Irish Times (13.05.2021),
We can look to Arthur Machen for the mystical (“Omnia exeunt in mysterium”), (…).
The phrase is also used as a chapter title in The Doctrine of the Trinity Apologetically Reconsidered by J. R. Illingworth (Macmillan, 1907). The phrase also occurs in that chapter (page 135):
All phenomena—matter, energy, electricity—are supposed to depend on this ether; yet it cannot itself be otherwise described than by symbolical terms, which even baffle our ordinary powers of conception, and may still, it is admitted, be remote from the truth. Here as elsewhere omnia exeunt in mysterium, as the schoolmen said of old,—“all things end in mystery,”—when we try to think them out.
In The Hidden Church of the Holy Graal by Arthur Edward Waite (1909), Book X, Chapter II: The Good Husbandsman:
The experiment had been pursued everywhere, the aphorism which ruled it being omnia exeunt in mysterium, the pursuit of that Mystery to which St. Augustine alluded when he said that Christianity had been always in the world--to which the New Testament itself alluded when speaking of the Lamb slain from the foundation of the cosmic order.
The phrase can also be found in a discussion of Ralph Waldo Emerson's poem The Sphinx:
And in “The Sphinx,” his expanding circles lead finally to the paradox quoted by Coleridge as one of the truths of Reason: “God is a circle, the centre of which is everywhere, and circumference nowhere.” Though all is riddle, we ared to the final, affirmative riddle: “Omnia exeunt in mysterium.”
From On Emerson: The Best from American Literature, edited by Edwin H. Cady and Louis J. Budd (Duke University Press, 1988, page 124.
The higher will must simply be accepted as a mystery that may be studied in its practical effects, but that, in its ultimate nature, is incapable of formulation. Herein the higher will is not peculiar. “All things,” according to the scholastic maxim, “end in a mystery” (Omnia exeunt in mysterium).
From Modern Criticism: Theory and Criticism edited by Walter E. Sutton and Richard Jackson Foster (The Odyssey Press, 1963; page 202). (See also “The Fallacy of Humanism” by Allen Tate, 1930.)
In effect they dramatize the Scholastic dictum Omnia exeunt in mysterium. Both works abandon the sleuthing reader to an inevitable yet unresolved circular self-detection within “a fog of mysterious theory”.
From The Ethos of Romance at the Turn of the Century by William J. Scheick (University of Texas Press, 2014).
In a footnote in a theological work:
According to Mackintosh, The Christian Apprehension of God, 140, “Of our thoughts concerning the spiritual being of God it is supremely true that omnia exeunt in mysterium—unfathomableness is the end of all.”
Quoted from Evangelical Calvinism: Essays Resourcing the Continuing Reformation of the Church edited by Myk Habets and Bobby Grow (2012).
These examples have two things in common. First, the quote is attributed to "the scholastics" without making that attribution more specific. (Even Arthur Edward Waite does not attribute the words to Augustine of Hippo.) Second, the word order is always the same, even though Latin word order was relatively free, which reinforces the impression that it was at some point a "dictum". But since searching for the literal phrase did not turn up any resources older than Coleridge, I also tried searches in which only "in mysterium" is in quotes, which should also give results where the verb "exeunt" is at the end (which was very common in Latin) instead of SVO order.
I specifically searched the works of Thomas Aquinas (one of the main figures of scholasticism; see Corpus Thomisticum) and Saint Augustine (because Augustianism was an important theological system before the arrival of Thomism; see augustinus.it and the Latin Library: Augustine of Hippo). This did not bring up any instance of the same words in the works of either.
A commentary on Revelation, 1:5-6 on Biblia.Work says,
Omnia exeunt in mysterium, says Sir Thomas Browne; all things issue in mystery.
This presumably refers to this Thomas Browne (1605–1682)), but I have not been able to find the quote in his writings. There is another Thomas Brown who was roughly a contemporary of Coleridge but I have not been able to find any writings by him.
The idea of mystery has not disappeared. For example, the document Theology Today: Perspectives, Principles and Criteria (2011) by the Vatican's International Theological Commission says (bold emphasis added):
The great diversity of matters that the theologian is led to consider finds its unity in this ultimate reference to God. All the ‘mysteries’ contained in diverse theological treatises refer to what is the single absolute Mystery in the strictest sense, namely, the Mystery of God. Reference to this Mystery unites theology, in the vast range of the latter’s subject matter and contexts, and the idea of reductio in Mysterium can be valuable as an expression of the dynamism that deeply unites theological propositions. Since the Mystery of God is revealed in Christ by the power of the Holy Spirit, Vatican II directed that all theological treatises ‘should be renewed through a more vivid contact with the Mystery of Christ and the history of salvation’.
Mystery is still important to the Catholic Church, as can also be seen in Part Two: The Celebration of the Christian Mystery in the Catechism of the Catholic Church. However, I am not aware that the phrase “omnia exeunt in mysterium” the views of the Pontifical Academy of Sciences, whose goals include
Promoting the progress of the mathematical, physical and natural sciences, and the study of related epistemological questions and issues.
Decadent and Occult Works by Arthur Machen, edited by Dennis Denisoff provides another clue. Machen's The Recluse of Bayswater (1895) contains the following passage:
I suspect that most modern chemists and biologists of repute would not hesitate to subscribe to the dictum of the old Schoolman, Omnia exeunt in mysterium, which means, I take it, that every branch of human knowledge if traced up to its source and final principles vanishes into mystery.
The sentence has a footnote saying,
In the Middle Ages, a Schooman was somebody who studied scholasticism, such as Thomas Aquinas (1225-1274). (…) The Latin phrase 'all things pass into mystery' is recognized as a dictum of Aquinas and other scholastic theologians.
In other words, it cannot definitively be attributed to a specific author.
Conclusion: the words “omnia exeunt in mysterium” (either in this word order or an alternative one) could not be traced back to any Church Father or any scholastic theologian, even though the spirit of the idea seems compatible with scholasticism.