There are several levels to the glow. You can see the (current) full progression path here - http://ffxiv.consolegameswiki.com/wiki/Zodiac_Weapons
It will start glowing once you get Zenith (see at the top of the page) for which you only need 3 Thavnarian Mists purchased from Auriana for 150 soldiery each, then take your weapon to the forge behind Gerolt.