I would put the burden of proof on her.
Most people do not think (rational) selfishness is moral, but they can easily understand WHY people want act in their own self-interest. The reason is pretty obvious to most people: their life and happiness demand it.
While they do not regard it as moral they do regard it as practical, to the extent they want to live and be happy. Most people want to live and be happy. (Unfortunately, this give rise to the moral-practical dichotomy. But that is an issue for another day.)
So at least implicitly, everybody knows why (rational) selfishness “makes sense”, even if they do not feel that they can morally justify it. After all, most people understand why people generally want a good job that pays well, or a good education, or have sex, or a nice house, or a nice car, or a hobby, or play video games, watch movies and listen to music, i.e., entertainment.
In fact, I would argue that most people would be baffled if you asked them: “Why is this good? Why is it good to pursue sex, money, housing or education? So what if it makes you happy? So what if it makes you enjoy life?”
No, the real question is and have always been: “WHY on earth, should we sacrifice ourselves? Why? Is there any rationally justifiable reason for it, at all?” So ask her questions such as: “Why should we give up our happiness - for the sake of others? Why should we sacrifice ourselves - for the sake of others? Why should we spend our days in misery - for the sake of others? Why? Why? Why? Give me just one good reason, one rational and logical reason.”
I would not be suprised if it turns out that most of her reasons will be illogical. (See Craig Biddle’s article “Altruism: The Morality of Logical Fallacies”.)
If you ever get her to question altruism, then make sure she understand the the essence of the alternative, that is of rational selfishness. How? Get her to read The Fountainhead and Atlas Shrugged.
If she is not willing to do that right away, then you should at least explain to her that a rational egoist lives by the guidance of his reason - not his whims; he thinks and acts long-term - not short-sighted; he regard honesty, justice and integrity as virtues; he do not believe that the happiness of one man demands any sacrifice - he believes that life and happiness demands a trade of values.
Thereby you clearly indicate that a rational egoist is NOT a S O B who would hurt others in their pursuit of their own happiness. This would also indicate that there is an alternative between sacrificing yourself to others and sacrificing others to yourself: pursuing your own happiness, by your own actions, without sacrificing anybody for anybody.
One final advice. Concretize, concretize, concretize. Give as many examples you can come up with. If you want to be sure the examples are good, then take them from Atlas Shrugged and The Fountainhead. Think of the characters in those novels. That should make even the most abstract aspects of the discussion easy to follow.
Good luck.