Your title is not helpful. It should contain words from your actual question, such as "me or my".
In the construction that you offered, both words are normally acceptable. The -ing word changes its status depending on the choice. With the possessive pronoun "my", it is a gerund (noun). With the objective pronoun "me", it is a participle (modifier).
Mom does not like my drinking on weekends. Mom does not like me drinking on weekends.