All that is needed is to synthesize an item at any idol. Also, it is wise to pay the gold to guarantee a successful synth.
Some helpful tips:
-Idols are the "merchants" of the game. You can purchase items, access item storage, synthesize, and sleep(which is how to progress character quests). There are 3 different Idols, however the different Idols have no bearing on synthesis.
That's all I got, it's been too long since I played to provide any relevant info. If you have anything to add feel free to post it in the comments and I'll add it in when I get chance.