That's a very good question.
In fact, even though it is technically correct,
Can you imagine his forgetting his own birthday?
sounds kind of silly, while
Can you imagine him forgetting his own birthday?
is absolutely grammatically correct.
The confusion (in the minds of those who compose such tests) stems, one would imagine, from the nominative vs dative-accusative concept.
Staunch proponents of perfect grammar have been scaring the less-grammatically-fortunate for decades into substituting "I" for "me" regardless of the case. Thanks to their efforts we now have such pearls as
It didn't matter to her and I.
Which is inconsistent, to say the least, since any case other than nominative that requires the speaker/writer to use the non-nominative form of personal pronouns ("her," "him," "them") would naturally and logically require using "me" rather than "I."
But: folks of humble origins and less-than-thorough education are now afraid to use "me," as well as "him" lest they be deemed "uncultured."
I'm tired of his criticizing everything!
though technically correct, sounds ... uh ... stupid ... or awkward, if you will, and
I'm tired of his criticism
is normally substituted, or, more frequently,
I'm tired of him criticizing everything
which flows a lot better, overtly colloquial though it may be.
The latter is what the authors of the test seemed to have had in mind when they urged the testees to avoid using "him" at all costs.