This page explains how to start and complete the Death Mountain's Secret side quest.
How to Start the Quest
Talk to Dugby in the Goron Hot Springs, which you can reach by following the road that goes northwest from Shae Mo'sah Shrine in Goron City, to start the quest.
How to Complete the Quest
Dugby hid a treasure between the Goron Hot Springs and the Bridge of Eldin. Go east, then climb onto the hills to the right of the path. Find the tallest part of this area and circle around it until you see cracked blocks. Use a bomb to blow them up, then get the Drillshaft in the cave there. Then go back and talk to Dugby to complete the quest.