Kind of a tricky question. If memory serves, Homer refers to the city of Athens by name in the Catalogue of Ships, and you can argue ad nauseam about the actual age of Homer in general or that passage in particular, but it’s certainly one of the oldest things in Greek literature. In any case I’d be happy to accept that the names of both the city and the goddess go back to the beginning of the ‘historical’ period.
Before that… well, there was almost certainly a Mycenaean citadel on the Acropolis in the Late Bronze Age, and people have drawn a connection here with the various jumbled stories about Erechtheus, the mythical king of Athens, living in Athena’s temple (the Mycenaeans didn’t have temples, as far as we can tell; religious and political authority were probably united under one roof). We don’t know what the Mycenaean city was called, though. I would bet good money it was something like the Classical “Athenai” because place names tend to be conservative, but we have no Linear B from Athens, so it’s not really possible to be airtight on that. One of the Knossos tablets does mention something called a-ta-na po-ti-ni-ja, and po-ti-ni-ja is a title commonly given to goddesses (the later Greek equivalent is potnia, which is one of the things Homer calls Athena), so possibly the reference is to a goddess called “Mistress Atana,” and just conceivably it might indicate that Athena is ultimately an aspect of the Minoan “great goddess” (and here much is made of the fact that snakes are sacred to Athena). The trouble is, these texts don’t actually tell us anything about the gods they mention; they’re just lists of the resources allocated to festivals and sacrifices and such, for accounting purposes. So we know that Athena probably exists and is worshipped in the Bronze Age in one form or another, but we don’t know whether she was anything like the Classical goddess, and we have no idea what, if any, connection she has to the city of Athens.