Is the wife who has not performed the obligatory Hajj more deserving of going with her husband than his mother who has performed it previously, especially given a prior promise to the wife to perform 'Umrah or Hajj?

No one is obligated to provide another person with the expenses for Hajj. A wife is entitled to her maintenance expenses if she travels with her husband's permission, and anything beyond that is from her own money, unless her husband donates it. Enabling someone who has not performed Hajj to do so is preferable to assisting another with a supererogatory act. The husband should explain the matter to his mother so that she departs with her full satisfaction towards him.