Favors were a perceived way to reward material acquisition in the Collection system, where you essentially have crafters 'throwing away' items and materials to get their scrips. They were to encourage players to go out and gather the materials they were going to use, by putting those Favor-items in, as along with the 'trash' stuff you gather, you were getting high gil-per-item materials as well.

Tuned well, they work, but the initial release (and even after wards with fixes) they were too much of a time sink to do what they were meant to.