From the Super Mario Wiki
The precious belonging is an object that belongs to a Toad in Mario & Luigi: Superstar Saga and its 3DS remake. In Toadtown Square, Mario can find this Toad walking up and down along a hedge. Upon talking to the Toad, Mario learns that the Toad has lost a precious belonging of his, and he asks Mario to help him find it. By walking south along the hedge, Mario can easily obtain this object. When he returns it to the Toad, Mario is given a 1-Up Mushroom.