It is generally believed that the founding of Athens predates the Mycenaean era. It is known that there were indeed prehistoric settlements in Attica, but exactly when the name "Athens" was first used for at least one of them is unknown.
According to Plato in the "Timaeus", Egyptian priests of Isis revealed to the visiting Solon that, according to their records, there was a prosperous city called "Athens" before 9600 BC. Of course, the accuracy of the report is ambiguous, as the calculation of the year too, but in the absence of more accurate dates and reports, it retains some indicative value.
The first inhabitants of the area are believed to be the Pelasgians. The first king of the city, according to mythology, was Kekropas during the 2nd or 3rd millennium BC, from whom the section between the Acropolis, Acharnes and Eleusis was named Kekropia. The inhabitants were Ionians who settled in Attica.
In the following video from Whiskytree company, which is behind the amazing effects in the movie Elysium, 5th century BC Athens is mapped out. The detail is so stunning that 1.62 billion items were used.