If my mother’s boyfriend of 10 years passed away, and now their family is trying to throw her out, what can she do? Does she have any rights?

You do not give us enough information for us to give a correct answer. Is she on the lease or ownership of the place if it is not a rental, has or does she pay any rent, who is trying to force her out [his estate being probated or the heirs just acting on their own] and on what basis, has she been given a three or thirty-sixty day notice, what promises did he make as to taking care of her [Lee Marvin style non-spousal agreement], which state are you in, why is she not writing in for information herself [she knows the facts better than you do and what results she wants], etc.

Without a written instrument such as a Will or Trust she has not acquired any property rights.