How much FFXIV Gil can I expect to spend purchasing the materials needed to create my relic weapon? It depends on how much you value your time versus your gil. I play on Cactuar, so you'll have to do your own market checking.Your going to need a:
Bloody Axe Head :(I believe that's the item name for Warriors, it drops from chests in Wanderer's Palace). These items were originally very pricey, then devalued as the first big wave of level 50's started farming Wanderer's, and now in some cases are rising again to the 10-20k price range since most have moved on to Amdapor.
2x Battledance III Materia:These are 30-50k right now but could vary on your server. If you have old tank quest gear from 45-50, try converting it and seeing if you can get some. You can try and farm this yourself while running Castrum (if you'd like to double dip efficiency and also make tokens) by wearing some 45+ DoW gear (preferably parry/tank gear for a better drop rate) or find a spot with densely packed mid-level mobs and go to town. You get spiritbond for kills not experience, so weaker mobs you can kill fast work well too.
I believe there is one or two additional mats for the crafting of the base item with the axe head, but those are pennies in comparison to the big ticket items (the axe head & materia). If you're patient though you can usually find someone to also craft/meld for tips which can save some additional money.