Broken Goron's Sword
First, Link must acquire the Poacher's Saw. He must then bring it to Mutoh, who is on the far side of the Gerudo Valley. If Link hasn't yet rescued the Carpenters, he will have to use either the Longshot or jump with Epona to get there, otherwise the bridge will be already fixed. Speaking to Mutoh will make him request his saw back, and in return he will give Link the Broken Goron's Sword. The next part is simple: Link has to bring the broken sword to Biggoron to fix it. He is located at the top of Death Mountain Trail, near the Death Mountain Crater entrance. If Link goes close enough to the left edge of the platform (which can be reached by climbing the wall after the path with the volcanic rocks falling) a mini-cutscene will be triggered, showing Biggoron standing up. Link will then be able to talk to him. Biggoron will have had his eyes "irrrrrrritated" by the recent eruption, and will not be able to repair the sword. Instead, he will take it for now, and give Link the Prescription for Eye Drops.
Note: If the Fire Temple has been completed and Volvagia defeated, the rocks will not fall on the path before the top of the trail; however, if it hasn't been done, rolling continuously is a very effective way to receive no damage, or very little.