The ownership of the house can depend on various factors, including the laws of the country or state in which the property is located and any legal agreements or arrangements made between the family members.
In many jurisdictions, if a family member builds a house on someone else's property, such as an elderly mother's property, without any formal agreement in place, the ownership of the house may still belong to the owner of the land, in this case, the elderly mother. This is because the principle of "land ownership" typically means that the person who owns the land also owns everything that is attached to it.
If the elderly mother passes away, the house she allowed her family member to build on her property may become part of her estate. The distribution of the estate, including the house, would then follow the legal process according to her will (if she had one) or the laws of intestate succession if there was no will.
However, it's essential to consult with a qualified attorney to get precise advice regarding the specific laws and regulations that apply in the relevant jurisdiction. The laws can vary significantly from one place to another, and individual circumstances can also play a significant role in determining the rightful ownership of the property after the elderly mother's passing.