At the time, I couldn't find the rule stated any earlier than William Cullen Bryant's (c. 1870) "Index Expurgatorius." But since then, I've found it in an earlier handbook, "Five Hundred Mistakes of Daily Occurrence in Speaking, Pronouncing, and Writing the English Language, Corrected," by one Walton Burgess (1855). "Over/more than" was mistake No. 130:
"There were not over twenty persons present:" say, more than. Such a use of this word is not frequent among writers of reputation. It may, however, be less improperly employed, where the sense invests it with more of a semblance to its literal signification: as, "This pair of chickens will weigh over seven pounds." Even in this case, it is better to say more than.Who was Walton Burgess? According to a New York Times obituary, he had worked for his father’s publishing business in New York City, then started his own. He “dropped dead of paralysis” in his office, aged 57, in December 1890. Since "Five Hundred Mistakes" was published by Daniel Burgess & Co., it sounds like the same W.B., but I found no clues to where he learned the "over" rule.
Not everyone subscribed to the rule, of course. In 1848, Seth T. Hurd used "over" in the forbidden way in his own "Grammatical Corrector, or, Vocabulary of the Common Errors of Speech," objecting to "A LONG MILE, for a little over, or a little more than a mile." And in 1900, Scott and Denney's "Elementary English Composition" said of "The buildings cost over a million dollars" simply: "This use is correct."
The OED shows "over" used with numbers since Old English. The "over/more than" thing is an invention of American journalism, unknown and unloved outside that shrinking cabal. In "Write It Right," I said that maybe only the death of newspapers would kill off this parasitic superstition; I hope that was just hyperbole.