A couple of thoughts. Both areas that have become more popular with Athenians in the last few years and have a lively, but more local, restaurant scene.
The first is Ano Petralona which is behind Philopappou, the hill across from the Acropolis. If you find somewhere in the north of Ano Petralona - on Ionon or Tritonos say - you can walk through to the Acropolis and major sites in 15 minutes. There's a nice walk up through the back of Philopappou into the Acropolis area. Five minutes the other way will take you to Ano Petralona metro which is handy if you want to go up to the Archaeological Museum.
If that doesn't suit you might look at Pangrati. This is across the National Gardens from Plaka, the main tourist area. If you stay near the Olympic Stadium you can just walk across. Alternatively get somewhere on or near Rozari and you can walk down to Evangelismos metro. This is one stop from the centre and on a direct line to the airport.
Have fun
Alan