I wouldn’t think of your taking the trouble.

(Grammar Questions I Couldn't Answer)

This question doesn't come from the students--it's actually a question I asked myself.

We're reading Mrs. Frisby and the Rats of NIMH in one of my teen classes, and while copying the text over from this website to a google docs (for the purposes of printing it out for the students), the google docs grammar checker flagged the "your" in this sentence as being ungrammatical.

‘Oh, I don’t know exactly,’ Jeremy said. ‘A couple of miles, I suppose.’
‘Mother, did you hear? You’ll be two miles up in the air.’
‘Martin, it won’t be necessary to go so high on this trip.’
Jeremy said cheerfully: ‘No, but I can, if you’d like.’
‘No, thank you. I wouldn’t think of your taking the trouble.’

So should it be "you're taking the trouble" ?
I Googled both "your taking the trouble" and "you're taking the trouble" and they both appear to be equally common.  So which form is correct?

My native speaker intuition is telling me that the sentence sounds perfectly natural.  But looking at the grammar, I think the rules are telling me it should be the accusative "you" here instead of "your".  I'm thinking "think of" is a phrasal verb, and that "you taking the trouble" would be the noun phrase following it, so the pronoun would normally take the accusative in this situation.  Do I have that right?
And I think that it would be obvious if we changed the pronouns, right?  e.g. "I wouldn't think of him taking the trouble"

So why is it common to say "your taking the trouble" ?