From what I understand, household gods were very common — personal traditions about a small god that perhaps your family worshipped for generations and had a small shrine to for protection. Some of these were picked up and developed into more general purposes for a larger audience.

The Romans in particular also had a habit of picking up local deities for a larger pantheon, in these cases less of choosing gods, it was more like reducing some gods with very elaborate lore down to just a few specific traits. Though of course some gods were invented by storytellers on the spot and picked up.