I propose the following. A new tab under Character > Achievement > Fame

Here you can spend the achievement points you've earned to gain traits. These traits are mainly there affects our luck rate in the world of Eorza.

These traits allow for the following suggestions OP has mentioned.

Open world.
Increases Monster Drop rate by 2%/4%/6%
Increase Monster Gil Drop rate by 2x
Increase Monster HQ Drop rate by 1%/2%/3%
Increase FATE Seal and Gil reward rate by 2x
Increase Leve Treasure Coffer Appearance Chance.
Increase Beast Tribe Allagan reward by 2x
Increase Treasure Hunt minion Reward rate by 2%
Increase Chance: Rare Retainer Ventures Rewards
Increase Gathering HQ rate success chance
Increase Crafting HQ rate success chance
Increase Rare Transmutation Materia success (Materia IV/V)

Dungeon, Raid, and Trials
Increase Rare Coffer Drop appearance: 2% chance of being rewarded the 2nd and 3rd treasure coffer in raids and trials. (Ex: Acheron's 3rd Coffer in CT and the 2nd and 3rd Coffer in Levi Extreme)
Increase Monster Gil Drop: .25x/.50x

PvP
???

There can also be other more unique implementation but for now this is the idea I propose that can tackle luck of low drop rate %.