The first instance in Google Books of "ran him over" in the sense "ran over him [with something heavy and dangerous]" is from 1936. From New Writing (London: Hogarth Press, Autumn 1936):
Rome was quite a different place in Benjamin's eyes now. In fact it was altogether transformed. There were plenty of people in the street, but no one took any notice of Benjamin. A dog barked furiously at him. A motor car nearly ran him over. A policeman regulating the traffic took him by the shoulder and gave him a violent push.
The next three occurrences are from the period 1941–1950. First from a summary of Lewis v. Jeffress, Court of Appeal of Louisiana (May 7, 1941), printed in Southern Reporter, second series (1941), and reproduced in Cyclopedia of Automobile Law and Practice: With Forms, Volume 9, Part 2 (1954):
A petition alleging that plaintiff was standing by a truck which had transported him to work when he was struck by defendant's automobile, and that defendant failed to straighten his automobile in making a sharp curve, and as a consequence struck plaintiff, knocked him to the ground, and ran him over, was insufficient, without more specific allegations of fact to show negligence, to justify a judgment against defendant, even if the allegations were admitted to be true. Rev. Civ. Code, art. 2315.
Ironically, the actual wording of the plaintiff's petition was
Your petitioner was standing by said truck on which he had been so transported to his work preparatory to removing some tools from said truck, when an automobile driven and operated by W.D. Jeffress, a resident of Jackson Parish, Louisiana, struck your petitioner and knocked him to the ground and ran over him, doing petitioner physical and bodily injuries as hereinafter set forth.
From McCall's, volume 71 (1943):
Then, there were Paval and Mike and Maria. And Nicolas. Here, her face saddened; the lips drooped over bad, stumpy teeth. "Nicolas, he is a cripple. A truck ran him over when he was a little boy. . . ."
And from The Horn Book (January–February 1950):
I have very sad news for you. Remember that boy you took to the puppet show with me and Irene at the public library? Well that boy went to buy some artificial flowers for a teacher in his school (sometime in February she sent him) and a truck ran him over. Isn't it sad?
So with regard to the source of these early instances of "ran him over" in the relevant sense, we have one instance from London (in 1936), one from Louisiana (in 1941), and two from national U.S. magazines (in 1943 and 1950). To me that suggests widely scattered informal use of the syntax, not especially localized Northern U.S. use.
The Library of Congress's Chronicling America newspaper database (which is searchable across the years 1836–1922) has a couple of considerably older matches for "ran him over." From C. Nolte, "A Trip to the Masai Steppes," in The Washington, D.C., National Tribune (January 4, 1900):
At the same moment that I started the rhinoceros selected one of my Somali soldiers—"Hassan Goulet"—ran him over and began to belabor him with his horn, which luckily was not greatly developed yet.
And from "Don't Always Work," in The [Chicago, Illinois] Day Book (June 23, 1913):
He found a horseshoe on the road
And likewise a four-leaf clover,
And as he stopped to pick them up
An auto ran him over.
These examples show that the wording "ran him over" in the sense of "trampled or crushed him" goes back more than a century in published writing.