Must a son carry out his mother's dying wish that he not separate from his wife, knowing that their marital relationship has been troubled for two years?

Your brother is not obligated to carry out his mother's will regarding not divorcing his wife, nor is he religiously bound to obey her in this matter unless there is a legitimate religious justification. Your brother should not leave his wife's situation in limbo, and he should strive for reconciliation as much as he can, for the continuity of marriage and the stability of the family are among the objectives of Islamic law. If the benefit lies in divorce, then he should separate with kindness, and perhaps Allah will ordain good for each of them, in fulfillment of His saying, the Almighty: "But if they separate [by divorce], Allah will enrich each of them from His abundance. And Allah is ever Encompassing and Wise." (An-Nisa: 130).