Cool question! While all Greeks knew the primary tales of Theseus, such as his battle with the Minotaur, his subsequent desertion of Ariadne, his abduction of Helen, and his journey to the Underworld, the Athenians in the late 6th century BCE began to add new tales to the legend of Theseus, and this is where he really becomes their mascot. Herakles was basically the hero for all Greeks, but the Athenians wanted a hero specifically for themselves.

So, as per the modern standard for things these days, they wrote a fanfiction. Of Herakles. His road to Athens was his version of the Labours, during which he apparently had similar Herculean strength, his weapon was a club (except his was iron, so it was clearly way better than Herakles. Clearly.), etc etc. They even added a little bit to tie him into the Iliad, making him the first guy who kidnapped Helen. Except that she was about 6, which is why he didn't actually get to marry her - was a bit too young for that.

EDIT: So both, combined with a sense of......."mascotness," I guess you could call it? He wasn't just a king, or a hero, or a monster-killer. He was their king, their hero, their monster-hunter.

Hope that answers your question! :) A great resource that I highly recommend if you're interested in learning more about stuff like this is Carpenter's Art and Myth in Ancient Greece. Really good book, and the artwork in it is some really fantastic stuff!