I'm sure by now it's become obvious there are essentially two forms of rebuttal to the Men's Rights Movement. The frontal assault, in which MRA's are characterized as Patriarchal Cretins out to enslave women "once again". The other is the "Many Feminists actually agree with you" argument, also known as the Patriarchy Hurts Men Too argument.
...So, how to respond?
In the first instance the response is rather obvious I would think... Attacking the messenger is only ever used as a distraction tactic, so simply asking for specifics as to what is wrong with your argument should suffice. This approach may lead to a "battle of the studies" type of debate...which is boring and no one wants to read those (which the Feminist agitator will be aware of). In that case, I would suggest keeping the conversation on "big picture" stuff (ie, why shouldn't men have Reproductive rights? Doesn't that mean that you then indeed DO advocate one set of rights for men, and another for women? Four legs good, two legs bad?).
The PHMTers, and the sister-group the "concerned Feminists", are tougher to deal with (and discern from genuine curiousity/concern). This umbrella also covers the NMBOF (not my brand of Feminism) approach in my view. For them, I would ask what Feminism has done, specifically, for men? If men's issues are "up to men to solve", what exactly qualifies THEM to even comment on men's issues, let alone claim expertise? Another qualifier would be their belief in Patriarchy Theory (do they think men oppressed women for thousands of years?) - a belief that requires the elimination of male power of all forms. This is man-hatred, and cannot be viewed otherwise.
The main thing to remember is that Patriarchy Theory (and the idea of historical oppression and the need for compensation) runs throughout Feminism. This will often be used to justify current injustices.
Of course, not believing in Patriarchy Theory(tm) essentially means your next question could easily be "How are you a Feminist then?" It's quite possible she's actually an MRA and didn't know it.
It seems almost too easy, but that's really all there is to it. Don't get bogged down in studies and statistics, appeal to logic and "common sense" whenever possible, don't be ashamed of saying your piece and don't hesitate to make sure it's heard, and insist that actions speak when words only promise.
A final note. The use of the "rape Card" is becoming more and more common, and it's destructive to the current thread, but I do not believe there should be much cause for concern here. It's already becoming sort of background noise, with the vast majority of those who comment on these subjects sick to death of the topic. But more importantly, this is an area where we can wedge a couple issues in. Prison Rape as consequence of imprisonment for CS debt,or false accusations in custody cases meaning innocent men railroaded into Prison being raped....well, how do you defend these things happening to good men, especially if you're "eager" to display that you really don't hate men....despite what 80% of the world thinks..
So you see, we can do much the same thing back....if we're so inclined..