Visit the official Athens tourism website for an overview of the city. These resources provide insight into the personality of the city, and its different neighborhoods. Using the tools provided on this site, you will be able to determine the best area for your stay.
Visit some apartment rental sites. Pay particular attention to sites that feature properties being rented by owners, and not brokers. Searching these sites, over the heavily marketed vacation rental sites, can save you a significant amount of money.
Search for properties just outside of the city center, or in the nearby port town of Pireaus. These properties tend to be lower in price. If you are seeking a budget option, these may fit the bill. Pireaus is popular amongst the major Mediterranean cruise lines, because it's an easy commute from the center of Athens.
Search some of the high-end broker sites, that specialize in luxury or landmark properties. If you are looking for a special accommodation, or an event space, these brokers can help you to find what you are looking for.