|The following is based on Fallout: Brotherhood of Steel and has not been confirmed by canon sources.|
Mary's mother died in the ruined area of the vault. If you find a ring there, return it to Mary in the garden.”— Quest description
After been given the ruins keycard by Patty, the Secret Vault - Ruins can be entered. Head west past the first door. Turn north when you get to the second one. Walk a ways until you see a door to the east. Enter it, take a quick north and then a west after you leave. The room you are entering after you turn west has Mary's mothers gold ring. Pick it up and head south out the next door.