How can my mother atone for taking money from someone for whom she was saving it without his knowledge, given that she cannot reach him now to return the money, and knowing that she was unaware of the prohibition of her action at the time?

Your mother's handling of the trust (amanah) without the owner's permission is Islamically unlawful (haram) and contrary to safeguarding the trust. If she did so out of ignorance, there is no sin upon her, provided she guarantees the money. She must keep the money until its owner comes or she searches for him. If she despairs of reaching him, she should give the equivalent amount in charity with the intention on behalf of its owner.