Ed, of course I have to guess about the use of your student's "Him driving
our car would scare me to death."  But I'm generally foolish enough to go
where no grammarian has gone before.
 
It COULD be that your student simply made a typo and meant to write:
 
         His driving our car would scare me to death.
 
After all, you almost always have the option of putting (leaving, if you
will) a genitive with a gerund.  For instance
 
     Julie's cooking made the guests ill.
     Dad's driving bothered Mom.
 
That seems to me the most likely explanation, though even in this revision
the NP object of the gerund, "our car," makes an awkward sentence.  One other
explanation might be (thank goodness for the conditional) that the student
meant:
 
     His driving of our car would scare me to death.
 
To my ear, the preposition between the gerund and its object makes the gerund
phrase more "acceptable," almost like
 
     The bombing of London caused great devastation.
 
I use the word "acceptable" with some trepidation.  I don't mean to sound
rigid and prescriptive.  But the the prhase without the preposition is
awkward; the preposition seems to make it a bit less awkward.
 
There are probably certain underlying structural rules about which subclasses
of verbs can be made gerunds in which way.  But such rules would be pretty
hard to state, I imagine.  Perhaps your student is not enough of a reader to
have discovered (intuited?) all the rules for gerund making, and he/she is
just awkward at it.   I'd compliment his/her attempt (after all, gerunds are
pretty sophisticated structures) and point out that sometimes you have
options in constructing gerund phrases, that perhaps the next time he/she
made one, he/she could try other versions.
 
I hope this is the kind of comment you were looking for, Ed.   --Max