Is it permissible to befriend a girl who has repented from prohibitions if the mother objects to it?

You have done well in disliking the girl who defied her Lord with sins, and in loving her after her repentance. This indicates the soundness of your standards and that your love and dislike are for the sake of Allah. If your mother insists on cutting off your relationship with your friend after her repentance, try to convince her that your friend has changed and seek help through supplication. If your mother does not respond, then it is obligatory for you to obey her, because obedience to parents is obligatory except in sin, and continuing the friendship is not obligatory. However, your obedience to your mother does not preclude other rights of friendship, such as the right of a Muslim over another Muslim.