There is an expression used when you invite, or give advice to, someone, "how about -ing?". Regarding this, my question is whether or not it's possible to add a genitive personal pronoun, just like the following expression:
Do you mind my/your opening the window?
For example, is the following example correct?
How about your going to see her in the hospital? (To recommend the hearer, not anyone else, seeing a woman in the hospital)
Thank you.
Your logic closely follows that of omitted "you" in imperative sentences, which can be re-introduced for emphasis:
I'll search upstairs, you search downstairs, and Tony, you search the basement.
I would say your example could be considered correct in the right context, such as the one you've described.
It is, however, irregular usage, and liable to be misheard. There's probably a better way to express the same idea.
(And of course, even if we argue that the subject was coded in the sentence all along, it's quite wrong to express it when emphasis isn't needed, because it changes the grammaticality of the sentence. Absent of context, "You bring me a pen" reads as an observation, and the noun "your going to see her" is a matter of fact.)
