I recall that in ancient Greece some gods were patrons of specific cities, and that these gods prominence reflected that of the city, but how these gods were created in the first place? Did an important person made a story up about an ancestor god to justify their power, like the whole Aeneas thing being the ancestor of Julius Caesar? We surely have first accounts of gods, do we know the specific process that brought them into "existence"?
We haven't stopped "creating" divine figures. Even in monotheistic religions you can create new divinities, or at least psuedo-divinities. In Christianity, specifically Catholicism, we see this in the canonization of new saints. These saints become symbols of protection and essentially take on the roles of minor, household or patron deities. Pope Francis has canonized 55 saints and while some won't gain much status, others like Mother Teresa were already being deified before their canonization.
The other modern example is Hinduism. Hinduism is an old set of religious beliefs and traditions, many of which are polytheistic of a kind. But it is not just an old, ancient set of practices, these traditions are living and we can see examples of the creation of new deities or traditions in the present. There are a few examples in more modern times that might help you.
One is Shirdi Sai Baba (1838-1918). He was originally a guru/fakir and gained a massive following while he was alive. During his life, and especially after, he became deified and is now worshiped by many. There are a few important notes here though. One is that many worship him not as a god, but as a guru and in a lot of Hindu traditions, you worship your guru. This is seen as showing respect to the person who is teaching and raising you. So for these people, he is not seen as a god, but an enlightened teacher. For others however, he is a god, either in his own right or as an incarnation of Dattatreya. He is also revered by Muslims as well in the capacity of a saint, similar to in Christianity.
The next is Swaminarayan (1781-1830). He was a yogi who taught that there was a Supreme Deity that had a physical manifestation (This still jives with having multiple gods. Hinduism is complicated). He is worshiped as an Earthly manifestation of that god.
There is also Kaival. Kaival is not a person, it is a conception of a Supreme Creator without form (Again, Hinduism is complicated). This idea was brought forth by Karunasagar in the late 18th century. Karunasagar is not worshiped as a god, but instead is thought of as a divine messenger. The followers say they pray to his image to use it as a crutch, because it is difficult to conceive or understand of a formless creator.
The last is Santoshi Maa. She was a goddess created in the 1960s and then later popularized by the movie Jai Santoshi Maa. She is a new goddess that has no real ties in older scripture, but has become massively popular.
I was gonna end this, but I realized that I do have ancient examples as well and I don't want to delete everything I wrote so time to go back to Ancient Greece.
Hermes is said to basically be the deified version of the herma, or rock road markers often with a phallus. One theory as pesented in "Woden, Hermes and Pushan" is that Hermes split off from Pan who was in turn a derivation of a PIE god. The theory says that Pan was originally a god of forests and travelers and as such he was associated with the herma and gained an epithet. That epithet split off to become Hermes and took most of his divine sphere with him.
I also know that Hades is not attested in the Mycenean inscriptions, but I know nothing about his origins.
Hope this helps a bit
There is the case of the Egyptian god Aten.
Originally, Aten was nothing more than the "disc" of the sun; a minor aspect of the Egyptian god Ra. This "disc" referred to anything flat and circular; the sun was called the "disc of the day" while the moon was called the "silver disc of the night". The first mention of Aten, as anything more than Ra's disc, was in ancient Egypt's 12th dynasty in which a deceased king rose to the heavens and merged with Him (ie: the story of Sinuhe).
It was only in the 18th dynasty, under the reigns of Amenhotep III and Amenhotep IV/Akhenaten that Aten rose to a prominent - supreme - role in Egyptian society. Amenhotep IV/Akhenaten even created a new city (the city of Akhetaten) to devote to Aten.
If you mean within the context of the religion itself, then yes. There's numerous cases in Greek mythology - like the birth of Athena or Hercules.