Athens has the Acropolis, the home temple of Athena Nike (victory). The myth goes that Athena and Poseidon each wanted Athens to pay homage to them; both wanted it so fiercely that it almost created a war between the two. As a contest to see who would become the patron, Athena proposed that both gods present the people of Athens with an offering, and when the preferred one was picked, that god would be the patron. Poseidon, with a mighty crash of his trident creates a body of water ( the sea now known as Erekhtheis). However, when inspected it was found to be salt water, which the people of Athens could not drink from. Athena, ever the strategist, planted an olive tree, from which the people of Athens could use the olives and the oil as food, and light their lamps, and from the wood they could built their boats and houses.

1 Huwit, Jeffrey H. The Athenian Acropolis: History, Mythology, and Archeology from the Neolithic Era to the Present. Cambridge: Cambridge University Press, 2000.

2 Apollodorus, "Bibliotheca 3. 14. 1" Trans. Sir James George Frazer. Loeb Classical Library Volumes, 121 & 122. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd., 1921.

3 Pseudo-Hyginus, "Fabulae 164." Trans. Mary Grant. University of Kansas Publications in Humanistic Studies, no. 34. Lawrence: University of Kansas Press, 1960.

Edit: Added some sources, once I am not on my phone I will properly format

EDIT2: I couldn't help myself; cross culturally there is also a tie between the founding o major Japanese temples and their respective bosatsu and gods. I don't have the text close at hand as it's currently in a box while I re-arrange all of my books, but through a noble princess, Amaterasu guided the Japanese to the site of Ise, where they were to build her major shrine an house the three sacred relics. I can't quite recall the pages on which the primary text is translated, but they book itself is called Sources of Japanese Tradition Volume One: From Earliest Times to 1600 eds. WM Theodore de Bary, Donald Keene, George Tanabe, and Paul Varley. 2nd Ed. New York: Columbia University Press, 2001. **

Like many other cultures, Greek religious worship centered around local cults, and the traditions of each cult were taken separately. Paphos was only one of several cult centers of Aphrodite which claimed to be her birthplace, or at least her home cult. The other most important site of Aphrodite's cult which claimed to be her birthplace was on Cythera, off the coast of the Peloponnese. Aphrodite, furthermore, is a rather difficult goddess, because more than most of the Greek gods she was an amalgam of countless fertility deities, which makes it extremely difficult to determine where the original centers of worship for the actual goddess Aphrodite were, as opposed to locations of other fertility cults which became associated with and absorbed by Aphrodite's. For example, the cult of Aphrodite of Ephesus is not actually a cult of Aphrodite at all, but rather a fertility goddess worshiped in parts of Asia Minor who was associated with Aphrodite and whose worship eventually became equated with the rituals of certain Ionian cults of Aphrodite.

Given that this is the case with many Greek gods, it's usually easier to refer to the home locations of various cults, each of which worships the god in a different aspect. In many of these cases this was originally a completely separate god, usually local (although not always--many of Zeus' various aspects, for example, seem to have been originally separate gods that were not tied to locations). Probably the easiest, and most classic, example is the two cults of Apollo. Traditionally Apollo had two competing cults, both of which were major centers of worship and both of which contained important oracles. The cult of Pythian Apollo at Delphi was a center mainly of Dorian worship, whereas the cult of Delian Apollo (on Delos, obviously) was mainly a center of Athenian and Ionian ritual worship (it's actually possible to trace to some degree the relative influence of the two sides on Greece as a whole by examining which cult was more important at any given moment). Zeus also had several different local cults--actually, most cities had their own cult of Zeus. Most traditions claimed that Zeus was raised on Mount Ida in Crete (not the one in Asia Minor, which was the seat of Cybele's cult), and there was a major temple there, although there were other competing locations for Zeus' birth. Much of Zeus' worship was actually unrestricted to local cults, since many of his aspects (such as his aspect as the Thunderer or the Oathkeeper) were not tied to localities. However, the most important cults of Zeus were not at Crete, but at Olympia and arguably in isolated Arcadia. The history behind the formation of the cult of Zeus at Olympia is pretty complicated and there are competing theories about it. It seems originally to have been some sort of Dorian cult center which grew over time, for reasons which are a bit beyond us. The cult of Zeus Lycaeus in Arcadia, probably one of the oldest cults remaining in the Classical Period, was a site of very bizarre worship, and all kinds of stories of cannibalism and werewolves floated around regarding its secretive and archaic practices.

There are a few gods that have very few cult centers. Athena, for example, is mainly restricted to Attica. Why this is is very uncertain. Athena is one of the oldest goddesses in Greek culture--she seems to be one of the few goddesses mentioned in the Linear B tablets, under the name Potnia Athena, Mistress of Athens. Most scholars believe that the story of Athens being named after Athena is backwards. Athena's classical name is actually Athenaia, and her Homeric name is Athene, and Athena is actually an abbreviation used by many tragedians because it fits into meter better. There's a lot of evidence to suggest that Athena was originally several different goddesses, and one plausible explanation is that somehow (possibly following the disturbances in the Peloponnese at the end of the Bronze Age) they all became equated with the local goddess of Athens--unfortunately there's not much evidence supporting or refuting this.

A few gods had no real ritual centers. Dionysus, for example, had no really central cults. Instead most cities had smaller cults to him in his various aspects, but there wasn't really a center of worship, although his worship was particularly associated with Asia Minor and Phrygia in particular, even though the cult centers there were no more central than anywhere else. Dionysus is a funny god. The Greeks believed his cult was a late introduction, but he's one of the few gods whose name is positively identified in the Linear B tablets. What exactly is going on with Dionysus' cults is pretty confusing and not entirely clear. Another similar god is Hermes, who interestingly often shared temple centers with Aphrodite. Hermes' major cult center is in Arcadia, but that's not really saying much since it really didn't matter much to anyone. In Hermes' case it's probably because he was an extremely local god, one who protected homes and public spaces. He doesn't really have a strong cult-center, therefore. After all, he's worshiped outside and inside people's homes, at crossroads, and generally wherever there's a herm.

I'm not sure if this really answers your question or if it's exactly what you're looking for, but I hope I helped a bit