The Biggoron’s Sword side quest involves a number of steps that eventually lead to a reward that is similar to the Giant’s Knife but is unbreakable. The 
Biggoron’s Sword is an extremely strong weapon and can be very helpful throughout the game. The steps for obtaining it are listed below.
The first steps can be completed once Link becomes an adult. Start off by speaking to the woman from the Cucco side quest that you completed in Kakariko Village to obtain the Bottles early on in the game – she is located on the southern side of Kakariko Village near the Cucco pen.
Speak to her as adult Link and she will provide you with a special type of egg (called a Pocket Egg) that can be used to hatch a Pocket Cucco. She asks that you take a Cucco along with you and allow it to “CUCCOOO” in order to wake someone up.
The Pocket Egg will need to hatch into a Cucco before you can proceed with the next step. You can wait and allow it to hatch naturally by allowing a full day to pass or you can use the Sun’s Song to speed up time and force the egg to hatch quicker. Use the Pocket Cucco to wake up Talon who can be found in the first house near the center of Kakariko Village (shown in the screenshot above).
Take the Pocket Cucco back to the woman who provided it to you – target her and then pull out the Cucco. She will then provide you with “an extremely valuable Cucco” named Cojiro.
This next step should wait until you have obtained Epona the Horse.
Head to the Kokiri Forest and then enter the Lost Woods. Take the first left to where the Skull Kid used to hang out. The creepy man from Kakariko Village is now sitting up against the tree stump.
Use the Blue Cucco, Cojiro, to wake the man up. The man will ask you to deliver some “stuff” to the old hag in the potion shop in Kakariko Village. Agree to take the “stuff” which ends up being an Odd Mushroom. You now have a time limit for taking the Odd Mushroom to the old hag.
This is a regular run for the most part, but you will have to make use of Epona out in Hyrule Field. You can also use the Magic Bean Plant to speedily make it to the bridge to exit the Lost Woods / Kokiri Forest. Just drop off of the Magic Bean Plant when it hovers over the bridge.
Ride back to Kakariko Village and enter the Potion Shop (the northernmost house on the right side, along the way to the Death Mountain Trail) and go through the passageway leading out back. Climb the steps and enter the red house to find another Potion Shop with the “old hag” inside. Use the video below to help guide you:
Head back to the Lost Woods to where the weird man was sitting. Instead of the man you will find a small girl. She will tell you that he is now lost and has become a Stalfos.
Use the Odd Potion near her (or while targeting her) and she will tell you that he left his saw and that you should give the medicine back to her. Agree to give it to her and she will provide you with the Poacher’s Saw.
Travel to Gerudo Valley and use Epona to jump across the gorge (or 
Longshot across) and give the Poacher’s Saw to the Master Carpenter standing outside of the tent. He will trade you the Poacher’s Saw for the Broken Goron’s Sword and will suggest that you visit Biggoron to get it repaired. The next steps of the side quest cannot be completed though until after you have completed the Ice Cavern later in the game.
The remaining steps can only be completed after the Ice Cavern dungeon.
Head up the Death Mountain Trail, up the cliff wall, and look off to the right side of the screen to find a huge Goron sitting there. Speak to him and he will tell you that he is the best when it comes to making blades. This Biggoron is involved in the Biggoron’s side quest and the final stages involve gathering and returning items to him.
Pull out the Broken Giant’s Knife near the Biggoron and he will tell you that, while he would like to repair it, his eyesight has left him due to an eruption at the volcano yesterday. Biggoron will provide you with a Prescription, the next item involved in the side quest, and will ask you to take it to King Zora.
Approach King Zora (after you have unfrozen him using Blue Fire from the Ice Cavern) and provide him with the Prescription. King Zora will read the Prescription and will provide Link with the ingredients to make the Eye Drops that Biggoron requires. The King will give Link the Eyeball Frog.
The Eyeball Frog needs to be transported to the Lakeside Laboratory at Lake Hylia as quickly as possible (3:00 minutes). Unfortunately you cannot use the Serenade of Water that Sheik just taught you to warp to Lake Hylia as a shortcut – you have to run there on foot.
Run down the stairway leading out of King Zora’s Chamber and follow the river bed down the Zora’s River. Hop on Epona and ride to Lake Hylia. Use the video below to help guide you:
The next step is to take the World’s Finest Eye Drops back to Biggoron. This run also involves a a time limit (4:00 minutes) and it pretty much requires you to have planted a Magic Bean Plant in the mound of soft soil in front of the Dodongo’s Cavern in order to make it up in time.
Ride Epona out of Lake Hylia and back to Kakariko Village. Run through to the Death Mountain Trail and then ride the Magic Bean Plant up to the top of Death Mountain.
Run towards the climbable wall and use either your Longshot or your 
Fairy Bow to kill all of the Skulltulas on the wall and then climb up to Biggoron. Use the video below to assist you:
Biggoron will provide you with a Claim Check to come back and claim the sword at a later date. You have to wait three days in total but you can use the Sun’s Song to speed the passage of time. Play the Sun’s Song 6 times to make 3 days go past and then present the Claim Check to Biggoron again to receive the Biggoron’s Sword as your reward.
The Biggoron’s Sword is much more powerful than the 
Master Sword but it requires two hands two wield. This means that Link will not be able to use his 
Hylian Shield while he has the Biggoron’s Sword equipped. The Biggoron’s Sword is extremely useful throughout the game.
