This one has long vexed me, and cropped up again this morning. Which is correct:
(a) I hope you don’t mind ME emailing you
(b) I hope you don’t mind MY emailing you.
I usually plump for (b) but I’m always unsure if this is correct … thanks for any help on this. 
Julie
b
I would have though (a) the correct usage, though (b) is heard in speech.
The two choices are different constructions, but they are both correct.
With the first choice (“me emailing you”), the direct object of the verb mind is the personal pronoun me, which is in the objective case; and emailing is a participle that is acting as an adjective modifying the pronoun.
With the second choice (“my emailing you”), the direct object of the verb mind is the gerund emailing, and my is a personal pronoun in the possessive case that is modifying the gerund.
Perhaps the confusion arises because a verb’s present participle and gerund are formed in the same way: [present stem] + -ing. Or perhaps because they are verbal forms that straddle more than one part of speech: a participle has characteristics of both a verb and an adjective, while a gerund has characteristics of both a verb and a noun. Likewise, a personal pronoun in the possessive case is a pronoun that acts like an adjective.
In this sentence, emailing is what’s being minded. For that reason, “my” modifies what’s being minded.
You could correctly express a similar idea with:
I hope you won’t mind me for emailing you.
In this revision, you’ve shifted what’s being minded from emailing to “me.” The difference is subtle, but in grammar it’s what matters.
And if you want to stretch the notion further, another rewrite might go:
I hope you won’t mind that I’m emailing you.
In each instance the “minding” has gone from
some emailing (that I contributed to)
to
me (for emailing)
to
the fact (that I am emailing you)
Again, subtle differences, each with a grammar rule to cover it.
I think the reason I usually go with (b) is because (a) is ambiguous - I could be asking if the sendee objects to the person (me) or the act (emailing).
Julie
Unless you are a Pirate.
“Yarrr! Shiver me timbers!”
I would use b, although a is not wrong as commented above. It depends on how you parse the sentence. But the usual parsing would have the pronoun be the subject of the verbal noun “emailing”, which is the gerund of a noun that has been verbed. As I mentioned in another thread the possessive is used for either the subject or the object of a verbal noun.
In the other structure, “me” is the direct object of “mind” and the subject of the gerund is what is technically called a trace (the ghost of an omitted word).
The answer is (b), because what is being said is, “I hope you won’t mind my act of emailing you.” The word emailing is a gerund, which is a verbal noun (can serve as both a noun and a verb in a sentence. So, to use it as a verb, you’ll need to use MY.
Did you read the rest of the thread, and brianmelendez’s reply, in particular? Both constructions are grammatically correct.
Personally, I wold use “me” if I wanted to sound informal, and “my” if I wanted to sound high falutin’.