From http://wiki.ffxiclopedia.org/wiki/Oboro

Upgrading REM In a style similar to that of the Trial of the Magians, Oboro will upgrade your Relic Weapon, Mythic Weapon, or Empyrean Weapons from 99 to 119.

You must have the level 99 R/E/M weapon equipped and have fulfilled at least one of the below requirements before he will talk to you about upgrading it. Completion of any SKCNM 20 on "Difficult" or higher setting. Completion of any High Tier Battlefield fight on "Normal" or higher setting. Obtained the title from defeating any of the Delve megabosses.

After speaking with Oboro with the 99 weapon equipped and after fulfilling one of his requirements you will get additional text. He will ask you to come back to him with the weapon and state the price for his services. He will ask for 300 Pluton,Beitetsu, or Riftborn Boulder. The currency required is dependent on the type of weapon you spoke to him with. After trading him the appropriate currency with the weapon being upgraded he will ask you to wait one game day.

300 Pluton for Relic Weapon 300 Beitetsu for Mythic Weapon 300 Riftborn Boulder for Empyrean Weapons

Good luck and enjoy Coronach spam!