From the Super Mario Wiki, the Mario encyclopedia
The Mushroom Handle is a special item in Paper Mario: The Origami King. It is found in a chest on Full Moon Island and is used to unlock the back door of the house on Mushroom Island. Mario must fish up a Sidestepper and defeat it to reveal a Magic Circle. This allows Olivia to become the Earth Vellumental. Not sure what to do, she decides to dance it out, raising the majority of the island out of the water to reveal a statue and taped treasure chest in the center.
Although the statue tells of the Diamond Key inside the chest, there is a simply the Mushroom Handle and a note from Luigi telling Mario to meet him at Mushroom Island. Olivia comments that it fits in Mario's hand well. Mario must take the handle back to Mushroom Island, thaw out the house if not done already, enter the house, and insert the handle into the matching door. Now able to open the door, they are able to enter the backyard and meet up with Luigi.