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Run him over and run over him have distinctly different nuances for me.

Run him over has clear and malicious intent (to damage with a vehicle)

Run over him, absurdly, admits the possibility that the incident may be for his own good/enjoyment, or at the very least an accident ... (a simple description of an action, without implied intent)

So, a 'bad guy' will try and run someone over. A 'good guy' may run over someone by mistake.

I am very surprised that you never heard this idiom 50 years ago. The OED has references going back to this one from 1860: "A carriage..darted under the arch of the gateway, almost running him over."

My suggestion is that it happened by association with other, similar verbs.

Mary knocked over the vase. Mary knocked it over.

Mary ran over the dog. Mary ran it over.

The word 'over' has different meanings in those two cases. However most people aren't linguists by nature. Most of us use language unthinkingly. If it 'sounds right' then we'll say it. The fact that a sequence of words may be illogical doesn't matter to most people as long as their friends can understand them.

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.

"Run over" is a verb with its own meaning, grammar, and pronunciation. You'll see the difference in pronunciation if you consider:

He ran over the grass.

which is ambiguous in writing, but not in pronunciation. If "over" has less stress than "ran", it specifies the place he ran or the path he took. If "over" has more stress than "ran", it means he affected the grass somehow, probably by driving a vehicle over it.

The difference in stress reflects a difference in constituent structure, and it follows the rule that the last stress in a constituent is strong. When "over" is part of the prepositional phrase "over the grass", the structure is

He ran [over the grass]

and "over" is not at the end of a constituent, so it has comparatively less stress. This is the interpretation meaning that the place he ran was over the grass. However, if "run over" is a verb, the constituent structure is

He [ran over] the grass

and now "over" is at the end of a constituent, the verb, so it has stronger stress.

The grammar is also different for the two structures. A prepositional phrase can be preposed in, for example, a relative clause:

The grass over which he ran was newly planted.

But this is not possible when "over" is part of the verb "run over".

There is nothing very unusual about the verb "run over". English has many such particle verbs, and they have been well studied. One of their special grammatical properties is that the particle can be moved to the right of a following object, and this movement is obligatory when the object is a pronoun.

The two versions appear to be accepted as correct usage now:

Run over (phrasal verb):

• 
  

  run over somebody | run over something | run somebody over | run something over

  
  
  
  
  
  • (of a vehicle or its driver) to knock a person or an animal down and drive over their body or a part of it.
    Two children were run over and killed.
    
  
  
  
  

Oxford Learners Dictionary

Ngram (run over him vs run him over) suggests that the 'bastardised' version actually became more popular from the late 60's but was used, though less , also before that date.

Donh't the two have different meanings? A car can run into you and knock you over, which means it ran you over. But unless the wheels actually went over you (uncommon nowadays) it did not run over you. It is certainly more common to talk of 'running over an animal' than 'running an animal over', because the result is a flattened body rather than something knocked flying.

This is somewhat speculative about the origins, but, if you compare English with Dutch you notice a distinct difference in how they prefix certain verbs with certain prepositions to form a totally new lexeme distinct from the original verb, often prefixing the verb with the preposition. For example, withholding something is literally saying holding it with you, but the lexeme has shifted to emphasize the with: the object is something that would normally be given but is now being reserved. Similar patterns seem to apply for the prefix out-, for example; to outfox someone is literally to "fox them out" but idiomatically the out- prefix suggests a totality to the action, as with outdoing yourself or outselling your competitors. Dutch goes crazy with this: an inbreaker is their term for a robber and an outfinder is their term for a discoverer (to convey the same meaning analogously in English you would say "breaker-inner" and "finder-outer"); to take something with you is the action known as meenemen or "with-taking.")

Along these lines, it is possible that we use the form ran him over in order to imply that he was overrun: though he generally did not travel under the car, nevertheless, when he and the car collided, the car had so much more power and force than he could ever hope to amass that he was simply destroyed.

The difference with a cat would be that usually the cat is small enough that it does pass underneath you.