This is a very subtle piece of wordplay, so it makes for an excellent question. The meaning, believe it or not, is God, and your answer "damn". This is analysed in An Ingenious Jest in Byron's "Don Juan", a paper by John I. Ades in Papers on Language and Literature 24(4) (1988), p. 446.
The Hebrew sacred name for God is YHVH, usually vocalized as "Yahweh" or "Yahveh." In Byron's day (as in many scholarly circles yet) it was thought that "Yahweh" derives from a form of the Hebrew verb "to be" used in Exodus 3:14-15, where, in reply to Moses's request for God's name, God replies, "I am". Thus, mutatis mutandis, the "I am" that Byron wittily asserts the English use to "govern d--n" is, of course, "God"; producing the mild oath, "goddamn." We see here the extraordinary lengths to which Byron would go to set up a jest. In stanza XIV Byron neatly accomplishes three objectives: he displays his etymological wit, he trivializes Donna Inez's (Lady Byron's) learning, and he fixes a satirical barb in the hide of the English gentry for habitually taking the Lord's name in vain.
In short: "the Hebrew noun which means 'I am'" is God, which is also used by the English to "govern damn" in the phrase god-damn. (Presumably it's damn, not darn as you guessed, to rhyme with "I am" in the previous line.)
The interpretation of YHVH as "I am" is supported by the English Standard Version of the Bible itself, as reproduced here:
The word Lord, when spelled with capital letters, stands for the divine name, YHWH, which is here connected with the verb hayah, “to be” in verse 14.