I had some reasons to think it was Leibniz. But years ago Tom Stone sent me an interesting note in which he said that Leonard Peikoff, in his History of Philosophy lecture series, had credited Antonius Andreas from the 12th century.
Here is my new clue. It's auto-translated from French, by Google, so the English is imperfect:
“The identity principle: Wundt says that “the law of identity was expressed for the first time in a pure logical form by Leibniz (Logik, T. II, p. 562)”. In fact, this one in proposed a great number of formulas, among which: “Each thing is what it is”, “A is A, B is B” (New Essays on Human Understanding, IV, 2, ed. Gehrardt, p. 343, sq.)… However Suarez already allotted to Antonius Andreas the following formula: Omne ens est ens, that it rejects besides like useless (Metaph., Disp., sect. III, n° 4).
If this is correct, Leibniz gets credit for "A is A". Antonius Andreas gets credit for "Omne ens est ens." (Which I think in idiomatic English might be close to "Each thing is itself".) Aristotle, it is true, had already touched on the issue.
French here. Google-translation here. It's from a French Scholastic philosophy site.
So, thank you Tom Stone for sending me that note - which I came across today - since I found this reference by Googling for Antonius Andreas and "a est a".
So it looks like Leibnitz was the first to say
A is A.
UPDATE: The Wundt mentioned is Wilhelm Wundt, and the citation seems to be to his book "Logik". The Suarez mentioned is Francisco Suarez, and the citation is to his Metaphysical Disputations.
2nd update: "Omne ens est ens" may be something more like "all being is being," or "whatever is, is". Wish I remembered my high school Latin!