If you are looking for a quiet area, and to avoid night noise, be careful, in the area around Syntagma Square you could have quiet pedestrian streets.
And noisy streets like:
Mitropoleos street, Adrianoy street, Ermou Street and Voulis street. Lots of traffic all the time (plus delivery service trucks and garbage collection). So noisy if you have a room facing these streets.
All these streets are west of Syntagma Square
Here are some other areas to avoid:
The immediate surroundings of Monastiraki Square, very lively until late in the evening:
The whole Psyri district which is excellent for going out in the evening but much less for sleeping:
Like Janet I confirm: Avoid all hotels on Leoforos Andrea Siggrou avenue, a large avenue with lots of traffic 24/24,
Also avoid all areas around Omonia Square where prices are cheaper but the environment is really not great for a first visit to the city.
I never stay in hotels when I'm in Athens, but here are 2 examples of well located and not overpriced hotels close to the main sites:
Herodion Hotel (already mentioned by janet)
Acropolis Hill Hotel (be careful, sloping street)