Final Fantasy XI answered this question via Trial of the Magians, allowing holders of Relic or Mythic weapons to upgrade their weapon appropriately -- and allowing everyone else to start a new weapon they can custom-tune. With the first level cap upgrade to the game coming in the June version update, it should come as no surprise that the Trial of the Magians system is getting a concurrent upgrade.
The fact that there will be new options to enhance and improve weapons almost goes without saying, as it lets players push their weaponry even further than before. There's also the promise of new trial varieties, and mention of being able to improve one's weapon skills themselves -- certainly an attractive proposition. If you're an adventurer on the go, you can also purchase a set of newly-added Magian Spectacles to check in on your trial status and progress along the upgrade path from anywhere. All told, the changes promise to improve on of the bigger new additions to Final Fantasy XI, whether you're swinging a Lumberjack or a Conqueror.