If you are lucky to know a local, or daring enough to drive yourself, it's definitely worth visiting Hymettos mountain, particularly its north-east side (drive up from the Katehaki or Papagou exits of the ring road - Attiki odos). Discover a Byzantine church, a traditional monastery and other archaeological sites of interest, have a drink in a cafe nearly hidden between the trees, enjoy some of the greatest panoramic views of Athens, do the bike or footpaths, or check for events/activities taking place throughout the year. Hymettos is also renowned for its marble and thyme honey, as well as for being Greece's most crazy mountain, (ie "trelos" or "trelovouno" in Greek) probably because of its propensity to change colour under shifting sunlight.
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