Both are correct. Would you mind me riding your bike tonight? (most common in modern English) ("me riding" is called a subjective gerund) Would you mind my riding your bike tonight? (less common - "traditional" English) ("my riding" is called a possessive gerund) This wikipedia article on gerunds has information including possessive and subjective forms: https://en.wikipedia.org/wiki/Gerund#Pattern_4b:_I_remember_her_coming.

Oh thanks a lot Simon~~

I would say that A is correct and B is incorrect. However you could hear someone using "my". For some odd reason it does not sound particularly strange to me in this case but "Would you mind my driving your car?" sounds wrong. I am not a teacher but I am a native speaker.