Testimonies, evidences, manuscripts, should be weighed not counted.

A textual criticism maxim that was surely understood back by the Reformation scholars. Thus they would have special discussions about the strengths or weaknesses of mss like Codex Vaticanus or Codex Bezae.

In the 1800s, Johan Nicolai Madvig (1804-1886) is given credit for applying the concept more precisely to classical scholarship.

The American (1886)
https://books.google.com/books?id=ACEgAQAAMAAJ&pg=PA231

Madvig acquired hia greatest reputation as a text-critic. He has with more clearness and precision than any other scholar laid the foundation for correct treatment of the old manuscripts. He has shown that manuscripts must be weighed, not counted, that one good manuscript has more value than scores of bad once.

A History of Classical Scholarship
John Edwin Sandys
https://books.google.com/books?id=16JUAAAAYAAJ&pg=PA321

From the outset of his career as a scholar, his special field had been verbal criticism. A rational method of estimating the value of mss, and applying the results, had lately come into vogue; mss were no longer to be counted, but to be weighed in comparison with the original archetype. This method was extended by Madvig, and was carried through with remarkable clearness and precision4. In the preface to the De Finibus there is a characteristic passage in which he compares the textual critic to a judge whose duty it is to elicit the truth from the conflict of evidence5.
Bengel is often given credit for being an early one, based on his setting up of manuscript families, although the phrase does not seem to appear.

Introduction to the textual criticism of the Greek New Testament (1901)
Eberhard Nestle
https://books.google.com/books?id=RRdVAAAAMAAJ&pg=PA17

...Bengel’s most important contribution to the textual criticism of the Greek New Testament consists in the sound critical principles which he laid down. He recognised that the witnesses must not be counted but weighed, i.e. classified, and he was accordingly the first to distinguish two great groups or families of manuscripts.
In 1848, John Scott Porter applies the phrase directly to New Testament "textual criticism".

Principles of Textual Criticism: With Their Application to the Old and New Testaments; Illustrated with Plates and Facsimiles of Biblical Documents,
https://books.google.com/books?id=NuxUAAAAcAAJ&pg=PA15

In endeavouring to disentangle the genuine readings from the mass thus collected, no one can imagine that we are to be influenced merely by the number of documents which favour or oppose any lection. Testimonies are to be weighed, not counted. The antiquity, the independence, the general fidelity, the prevailing character of each witness are to be taken into account.
John William Burgon showed how easily it could be misused and abused in New Testament Studies, noting the irrationality of the approaches of men like Tregelles and Westcott and Hort.

§ 2. Number.
II. We must proceed now to consider the other Notes, or Tests: and the next is Number.

1. That ‘ witnesses arc to be weighed—not counted,’— is a maxim of which we hear constantly. It may be said to embody much fundamental fallacy.

2. It assumes that the ‘ witnesses ’ we possess,— meaning thereby every single Codex, Version, Father—, (i) are capable of being weighed: and (2) that every individual Critic is competent to weigh them : neither of which propositions is true.

3- In the very form of the maxim,—‘ Not to be counted— but to be weighed,’—the undeniable fact is overlooked that ‘ number ’ is the most ordinary ingredient of weight, and indeed in matters of human testimony, is an element which even cannot be cast away. Ask one of Her Majesty’s Judges if it be not so. Ten witnesses (suppose) are called in to give evidence : of whom one resolutely contradicts what is solemnly deposed to by the other nine. Which of the two parties do we suppose the Judge will be inclined to believe ?

4. But it may be urged—would not the discovery of the one original autograph of the Gospels exceed in ‘ weight ’ any ‘ number ’ of copies which can be named ? No doubt it would, I answer. But only because it would be the original document, and not ‘a copy’ at all : not 'a witness’ to the fact, but the very fact itself. It would be as if in the midst of a trial,—turning, suppose, on the history of the will of some testator—, the dead man himself were to step into Court, and proclaim what had actually taken place. Yet the laws of Evidence would remain unchanged : and in the very next trial which came on, if one or two witnesses out of as many hundred were to claim that their evidence should be held to outweigh that of all the rest, they would be required to establish the reasonableness of their claim to the satisfaction of the Judge : or they must submit to the inevitable consequence of being left in an inconsiderable minority.

5. Number then constitutes Weight, or in other words,— since I have used ‘ Weight ’ here in a more general sense than usual,—is a Note of Truth. Not of course absolutely, as being the sole Test, but caeteris paribus, and in its own place and proportion. And this, happily, our opponents freely admit: so freely in fact, that my only wonder is that they do not discover their own inconsistency.

6. But the axiom in question labours under the far graver defect of disparaging the Divine method, under which in the multitude of evidence preserved all down the ages provision has been made as matter of hard fact, not by weight but by number, for the integrity of the Deposit. The prevalent use of the Holy Scriptures in the Church caused copies of them to abound everywhere. The demand enforced the supply. They were read in the public Services of the Church. The constant quotation of them by Ecclesiastical Writers from the first proves that they were a source to Christians of continual study, and that they were used as an ultimate appeal in the decision of knotty questions. They were cited copiously in Sermons. They were employed in the conversion of the heathen, and as in the case of St. Cyprian must have exercised a strong influence in bringing people to believe.

(continues, including Tregelles followed by Westcott and Hort.)
The always obtuse James Price tries to accuse Burgon, since he is really a Critical Text apologist behind his NKJV front.

King James Onlyism: A New Sect (2006)
James D. Price
https://books.google.com/books?id=hL4XgUSGP8sC&pg=PA232

And the maxim is often used through Aland or Metzger, as you can see here by Ronald J. Gordon.
https://www.cob-net.org/compare_greektext.php

So far, I have not found Hort actually using the phrase.

Maurice Robinson gives some history of its usage.

“It’s All About Variants”—Unless “No Longer Written”
—Maurice Robinson
https://books.google.com/books?id=cmZUDgAAQBAJ&pg=PA121

Even when “manuscripts must be weighed rather than counted”—a frequently repeated maxim that is applicable to a certain degree22—...

22. Wettlaufer, No Longer Written 18: “The single most important principle of modern textual criticism is that manuscripts must be weighed not counted”; restated at 33, 117.

The same principle is repeatedly stated in numerous sources, e.g.,

Aland and Aland, Text of the New Testament, 280;
Metzger, Textual Commentary, 12*;
Clarke, Textual Optimism, 37, 44, 51, 162, 174;
Wegner, Textual Criticism, 132, 212, 240, 248;
Holmes, “Reconstructing the Text,” 83-84;
Mounce, Greek for the Rest of Us 266.
When there is a spot more time, some of these, and others I have noticed, could be gone over.