Not only is this wrong on many levels, it is also illegal.
This kind of discussion falls under the FERPA law of privacy, I believe. In PA it is required that we have an updated inservice on this every year before the school year begins. We have been told in no uncertain terms to never discuss a student with any other teacher who has no reason to know information about said student, even if the child's name is not used. This includes in faculty rooms, or any casual converstion, such as a random meeting with another teacher at the grocery store, etc. As told by our superintendant, you never know who is around listening and may put 2 & 2 together to know who is the subject of the conversation.
Here is what I recommend:
1. by all means, write the newspaper article, but do not publish. She should write down in non-emotional terms exactly what happened. Time, place, persons involved, word-for-word repetition of what was said. Then add a paragraph about how it made her feel.
2. Parents should request a conference with the main teacher involved. Read the letter word for word. Expect at least an apology. If your cousin did not do well on the make-up b/c of the circumstances, request a retake.
3. If no satisfaction is provided at this meeting, go to the principal for resolution. The key here is to go up the chain of command. If you go right to the superintendant, their first question will be whether you spoke to the teacher and principal about it. If the answer is no, s/he will kick it right back there.
My husband and I are fortunate enough to both teach in the same school and often, due to our subject matter, have many of the same students. We talk about the students in our car or our home, where it is guaranteed that no one is around who will know the student. One year not too long ago, we found a girl who had become anorexic and cut herself, based on behavior changes that we had noted from the previous year. Her other teachers had only known her for a couple of months and did not realize the difference in her from the year before. We discussed the situation at length in our home (outside the district), then referred her to the proper counselors for help. Today she is a much happier, healthier student/
but again, the conversation was private, with no chance of it being overheard by the wrong persons.
If you get no resolution by the time you get to the super level, then let them know that your lawyer will be in touch. Nothing makes a district scramble to resolve a situation faster than threat of a lawsuit. Bear in mind that this could get the teacher fired. This may or may not be the right thing to happen to the teacher, but my guess is that the situation will be resolved before this point.
Good luck and keep up posted!