What is the ruling on dipping one's feet in a basin of water for ablution without rubbing them due to the difficulty of bending over to wash them?

Massaging the limbs during ablution (wudu') is a Sunnah (definition) according to the majority of jurists and not an obligation, and this is the relied-upon opinion. The Malikis, however, view it as an obligation, but it is waived in case of inability. Your mother's ablution is valid, and you should take on the task of massaging her feet.