I heard somewhere that secondary mission (grimoire/tome) collection improves the Emperor’s Gift, but since there is no in-game feedback (a la VT2) and the variance in overall gift quality has such a high deviation there is no way to really tell.
Edit: Looking at Patch #4 - Blessings of the Omnissiah, looks like it’s conditions that increase the “rarity” of items, but again, there is no feedback about it doing anything.