Is there a sound Hadith from the Prophet, peace and blessings be upon him, regarding writing "Bismillah Al-Rahman Al-Rahim" 130 times on the first of Muharram to bring happiness and ward off evil, or is this act an innovation?

What your mother read has no basis indicating its legality; it is an innovation. Goodness lies in following the Sunnah of the Prophet, peace and blessings be upon him, and evil lies in innovation in religion, as he, peace and blessings be upon him, said: "Whoever introduces into this matter of ours that which is not from it, it is to be rejected."