What is the ruling on me giving gold that was gifted to me to one of my needy friends without my mother's knowledge, knowing that she considers everything gifted to me as rightfully hers? And how can I rectify this mistake?

What the mother believes—that a gift or present to a child is the property of both parents—is incorrect. A gift or present is the property of the child, who may dispose of it once he or she reaches maturity and sound judgment. Your disposal of the gold is permissible, but it would have been better to choose a method that avoids suspicion and considers your mother's feelings. Informing her now about what happened might be best to gain her satisfaction. There is no sin upon you for this action.