Yeah heaven's eye III or quickarm III is the most logical. heaven's eye is the same as bard but no other class, quickarm isn't used by anyone. Every other is used by 2 classes already, plus there's kind of a logic(the crit are for both casters, the det are for both str melees, the parry for both tanks, the spellspeed for both healers), so heaven's eye due to both dex or a new set makes most sense. I bought a few quickarms and heaven's eye in preparation like 2weeks ago.
Things of note about the relic, it's alchemy crafted(and melded obviously) and you need Bloody Knives, which assuming they follow the pattern, will drop in the old ass wanderer's palace(where other bloody weapons for relics drop, other than whm). Could drop in one of the new dungeons instead though, guess we'll need to see. The other mats are garbage shit, one ochu vine(easy to farm if the prices skyrocket for whatever reason, east shroud in larkscall for example), one desert saffron(mid level herb so also easy) and one rosewood lumber, plus a bunch of crystals and what not.
However, you can get a 110weapon instantly by buying it after patch day for 1300soldiery, an UAT and a sands of time though(last 2 drop in ST which won't have a lock next patch so a few runs should net you both). The only reason not to do this is that Soldiery might turn into solid money from the new mats, but that's still several steps ahead in the relic stuff, and there's the Novus step which is still going to be super expensive shit. Assuming you use shit materias on that step though, Nexus is 115 and is going to be the best weapon unless you do 3rd coil, until whatever patch they make that adds the new drop for the 120/130 weapon to something else, which might not be before 2.5. 115 with garbage random stats vs 110 with potentially good stats is fairly close though.