Add a unique stat to the new Relics, so they are worth obtaining. Enough is enough.

[DragonSlayer45]

Can someone give me an example of a unique trait on a weapon that would not be useless or unbalanced to the ilvl scheme?

There's a ton of ways they can go about it without making it hideously unbalanced, just a boost to make the relic feel unique. Make the relic tie in with the job.

Dragoon relics can boost Jump damage.
Monk relics extend Greased Lightning duration.
Bard relics decrease the amount of MP songs use and expands the area of effect for songs.
Black Mage relics boost Astral Fire damage and also decrease the amount of MP used during Astral Fire.
White Mage relics boost outgoing healing and Holy damage.
Tank relics provide a decrease in damage.
Machinist relics decrease the cooldown time on Reload.

If they can't do it because "Item Level" then that shows how broken and bland the system really is.