What is the ruling on taking a sum of money that came as a result of deceiving others into believing in a debt that does not actually exist, and the mother refuses to return it or disclose the truth?

This money given by the person to your mother is not permissible for her, unless she informs him that she is not indebted and he agrees to give her the money. Otherwise, it is not permissible for her, because he gave her the money to repay a debt based on her inquiry about the debt amount. Thus, it is not permissible for her as long as she is not indebted. It is mentioned in Nihayat al-Muhtaj and Hashiyat al-Shirwani that whoever takes money based on the giver's assumption of a non-existent condition does not own it, because he seized it without the owner's consent. Therefore, the mother should be advised gently. If she insists on taking the money without informing its owner of the true situation, then it is unlawful money. If she has other lawful money, then her money is mixed, and there is no harm in spending from it. However, if she has no other money but this, then it is not permissible for you.