Aether is named after the Greek personification of the upper sky Aether may have had shrines in ancient Greece, but he had no know temples and it is unlikely he had a cult. The name was most likely chosen because its function was to Detoxify of Earth’s atmosphere.
Apollo is named after the Greek god of archery, music and dance, truth and prophecy, healing and diseases, the Sun and light, poetry, and more. The name was most likely chosen because of the truth and prophecy aspects of the god being its function was to educate new generations of humans created in Zero Dawn Cradle facilities.
Artimis is named after the Greek goddess of wild animals and the hunt which is why the Gaia function for Reintroduction of Earth’s faunal species from preserved genetic stocks would be named after her.
Demeter is named after the the Greek goddess of nature, the harvest, spring, summer, as well as other similar things. The harvest and nature aspects are why the name was chosen for the function that Reintroduces Earth’s floral species from preserved genetic stocks.
Eleuthla was the name of the goddess of childbirth and labor pains this is why the function for Reintroduction of the human species via cloning and raising of humans from genetic stocks at Cradle facilities.
Hades is the god of Hades; the underworld and the god have the same name. It is important to remember that the underworld is not hell but rather the place where all the souls would go, good or bad. Because of the popularity of Christianity Hades has been associated with Satan when in truth the two are quite different and Hades is not an evil god at all. None of the Greek deities were good or bad and were flawed. The Gaia function was named after him due to the death aspects of his character, but I feel it important to note that Hades went rogue due to an outside virus.
Hephestus is the Greek god of the forge, and the function does exactly that: forging Zero Dawn terraforming machines at Cauldrons.
Minerva is the roman counterpart to Athena goddess of wisdom and war. The Gaia function was used to create Faro Plague deactivation codes and construct transmission towers to broadcast Plague shutdown codes.
Poseidon is the Greek god of the sea and most other bodies of water. The Gaia function detoxified these bodies of water.