Do you mind one's doing sth ? 中,为什么要用one's

mind的用法是 mind doing sth ， 动名词做宾语。
这里用my 做 doing sth 的逻辑主语，应该用形容词性的物主代词，而不用me这个宾格，这里不要要求mind带宾语。
假如是类似这样的句型，就可以用me, want sb to do sth, ask sb to do sth,这里sb就是宾格。

1.mind doing something (mind做不及物动词使用)
Did you mind being away from home for so long?

2.mind somebody doing something (mind做及物动词使用)
Don't your parents mind you staying out so late?

应该都是可以的，my就是符合1的用法，me就是符合2的用法

－Would you mind my closing the window?

-Would you mind me opening that door?
-I'm sorry, but it's not allowed.

PS:
Do (Would) you mind doing sth.? 这一句型通常用来表示请他人做某事，常译为\"可否请你做......\"或\"劳驾你做......\"，多用Would开头，用Do的情况较少。