Lefteri's terrace is the best! I spent 3 weeks there, and I can tell you that "cozy" is an understatement! The bed outside is amazing, and I loved the "open air" living, it feels a bit like camping. On Tuesdays there is a street market downstairs, and a lot of cafes to pick around. Also, the Acropolis hill is really 10 minutes away, and even if you are in the center of Athens, it all has a very pleasant and relaxed vibe. I used to walk everywhere, almost no need for metro and buses! Wil definitely come back if in Athens again!