Insisting on a genitive pronoun (my, his) + noun in such constructions would be sheer pedantry, but in practice we do overwhelmingly prefer it to the simple pronoun (me, him) + verb...
It's worth homing in on that "flatline" above...
What those charts show is that although there's still a marked preference for the genitive, the usage is in fact declining, whereas the simple pronoun + verb form has gained traction.
Also note that if a "standard" noun can follow the genitive (in the event of his death) that's what we prefer. But we'd still rather stick to the genitive even with a gerund (in the event of his dying). Relatively speaking, the pronoun + verb form (in the event of him dying) isn't a popular choice.
Having said all that, the preference for a genitive/possessive + [gerund] noun form only really applies with pronouns. Consider this earlier question, and these results from Google Books...
in the event of the king being (3,010 hits)
in the event of the king's being (7 hits)
in the event of the king dying (1,520 hits)
in the event of the king's dying (114 hits)
Although as pointed out, with 21,700 hits, in the event of the king's death is far more common than either of those last two.
TL;DR: The genitive form is still far more common with pronouns - but it's becoming less so, and grammatically speaking there's nothing wrong with the [pro]noun + verb alternative. Kudos to @tchrist's comment below for guiding me to this from CGEL..
Modern usage manuals generally do not condemn non-genitives altogether (as Fowler did in early work), though they vary in their tolerance of them, the more conservative ones advocating a genitive except where it sounds awkward, stilted, or pedantic [...]