Oh dear. I fear your cousin is going to have a difficult life. Like other people who hold very strange beliefs that are not shared by 99.999% of the population, she will occasionally find herself participating in a conversation where the subject her beliefs pertain to comes up, and then she will be faced with the following choices about how to behave:
Pretending to go along with the premise assumed by the rest of society (e.g., that universities are good and socially beneficial institutions, that Elvis was not in fact abducted by aliens, etc). In the context of your question that means telling people where she went to school.
Staying true to her beliefs and responding honestly to any questions on the topic being discussed, which means she will have to explain her beliefs and accept that this will lead to the vast majority of people likely thinking she is an extremely unreasonable or even delusional person.
Somehow avoiding the discussion entirely by making up an excuse to stay out of the conversation and avoid answering related questions ("for personal reasons I no longer take part in discussions about Elvis").
The problem is that choice number 1 minimizes negative social consequences for her, but at the cost of making her feel like she is living a lie and betraying her beliefs, which she may view as an unbearable sacrifice to make. Choice number 2 will incur (as she may have already learned, or will likely learn very soon if she adopts this policy) very severe negative social consequences. And choice number 3 is also likely to make her appear to be a bit of a weirdo and incur negative social consequences (though probably less negative than with choice number 2), while at the same time giving her some of the same negative feelings that she is a coward for not defending her beliefs (similarly to choice number 1, although I expect the negative feelings will be less strong since she will avoid the subject rather than be actively complicit in denying her beliefs).
The bottom line is that the choice is entirely hers and she needs to think about what matters most to her in view of the above analysis. Unfortunately I don't see any solution that avoids negative feelings and her having a difficult time in certain social and professional interactions, as I said. My personal recommendation to her would be to re-examine her belief about universities -- obviously if she changes her mind about her misguided belief that universities are corrupt, immoral and oppressive institutions then the whole dilemma will disappear, which will be the happiest outcome. But I recognize that this may not be a very helpful suggestion since it is not so easy for someone to change a sincere and deeply held belief just because it is in their interest to do so.