Is it permissible for my mother to perform Hajj with money she received as a wedding gift (nuqūṭ) from my sister's wedding, after she had spent her own money on the wedding and then recovered it from the nuqūṭ?

If the custom is that the money gifted at a wedding goes to the mother, then it is hers to dispose of as she wishes, such as for Hajj and Umrah. If the custom is that it belongs to the daughter, then the mother is not allowed to take it, unless she intends to reclaim what she spent on the daughter's needs and takes only the amount she spent, or if the daughter donates it to her and grants her permission to use it. In such cases, there is no harm for her to perform Hajj or Umrah with it.