As a sidequest during the game, Henry can try to find this sword, which is hidden in a chest in a secret room in the airship. If this task is completed the player gets a medal.
Finding the Sword
Finding the Biggoron's sword requires you to play multiple scenarios of the game in the right order.
First Henry has to go through the Pure Blooded Thief ending, using the Grapple Gun to enter the Airship. He arrives in a computer room, where there is a gold key next to the computer. By clicking the key, Henry takes it along. Later, when he reaches the cells, Henry uses the key to free Dave the imprisoned security guard. When Henry breaks into the vault, Dave also briefly enters the Vault and drops a flash drive, which Henry can pick up. After the Airship crashes in the desert and the Center for Chaos Containment decides to step in, the flash drive can be seen in the drawer with the four disks Wilson can choose from. Clicking the flash drive reveals a code, 1-2-1-4.
When the game is finished, Henry has to go through the Government Supported Private Investigator ending, choosing the Ear Piece in the start menu. When Henry enters the ship by vacuuming an enemy and dropping down, there is a keypad on the wall. By clicking it, Henry can enter the code he retrieved from the flash drive, which will open a door somewhere else in the ship.
Finally, Henry has to go through the Rapidly Promoted Executive/Ruthless Bounty Hunter ending by choosing the cannon ball. When Henry starts chasing the Toppat Leader through a hallway, he comes across the now open door. Curious, he will enter it and find the Biggoron's Sword in a chest.
Fleeing The Complex
It can be assumed that the sword makes a return in Fleeing The Complex, in the scene where Henry fights Carlov in a sword fight.