Is it permissible for my mother to fast the vowed days—which are ten days—on Mondays and Thursdays only, knowing that she used to prefer fasting them, and is a single intention sufficient for fasting the vow, Monday, and Thursday?

Your mother must fulfill her vow in the manner in which she vowed it, because a conditional vow must be fulfilled if its condition is met. If she did not intend to fast the ten days consecutively, she may fast them on Mondays and Thursdays. By doing so, she will fulfill her vow and receive the reward for fasting on Mondays and Thursdays. And whoever fasts the day of Arafah or Ashura while owing make-up fasts from Ramadan (definition), his fast is valid. If he intends to fast this day as a make-up for Ramadan, he will receive both rewards.