There's a hidden stat based on which deity you chose at character creation.

As we know, there are divisions by bell, moons and years.

http://ffxiv.gamerescape.com/wiki/Time

The base drop rate is about 2%, however during a bell matching your deities element (4 periods of 1 game hour per day) you gain a 5% increased chance of a drop, making the rate around 7%. This is a very simple part of the equation.

Next there are moons, this is a boost of 20% for days matching your deities elements but factored by the phase of the moon. So a full moon on a day matching your entities element would give the full boost, but a half moon only a 10% boost, and a new moon 0%.

There's a boost during the exact month of your deity, but this is small, only around 2%.

There's a mega-bonanza birthday bonus where on the nameday of your character, you will get a fixed 40% drop rate.

But all of this only applies if you face the direction of your deities element whilst the FATE reward is calculated.