"He wants some weak-sounding sword called the 'Omega Weapon', whatever THAT is..."
-Gilgamesh, about the Landlord

The Omega Weapon is... um... well, actually, no one knows what it is. Gilgamesh claims it's a sword, which is probably why the Landlord would charge the weapon-hunter with the task of finding it. But a sword called the Omega Weapon does not exist in the Final Fantasy universe. There does, however, exist an extremely powerful monster called the Omega Weapon, and it is arguably the most powerful creature in existence. Gilgamesh may have assumed the Landlord was speaking of a sword due to the word "Weapon" being in the title. However, evidence for either claim does not yet exist. The reason why the Landlord wants it is also unknown.

It was revealed in Episode 0042 that Gilgamesh had succeeded in obtaining the Omega Weapon, but the details are all unknown.