A PhD student I know was terminated because of mental illness without any accommodation. What is the possibility of legal action?
As others have said, this is probably a question for a lawyer. That said, we don’t have any idea of what kind of information  the advisor/university/HR dept has. Perhaps they had information that made them think they had just cause. It could be that your relative is either ignoring or perhaps just overlooking something you don’t know. 

Depending on the environment, mental health issues could be a cause for termination because it would make you unfit or ineligible to continue. If the research involves very dangerous products (infectious organisms, highly lethal compounds) or a strict environment (BSL4, secure environment such as a government facility), the terms were probably spelled out somewhere in a document your relative signed.