You should try to visit the Kessarianí Monastery. This place has a lot of Byzantine art that you might enjoy. It's a really beautiful area and not packed with tourists. 

A great place to relax would be at the National Gardens. Have a picnic here, it was peaceful and different option as opposed to going and dining at a restaurant. 

If you're looking for fun restaurants, head to Petralona. Small town but beautiful and picturesque.

Athens is a cool city but I think you would find more unusual things outside of the city like on the islands! Mykonos is notoriously known as the party island, you could definitely find unique things to do there! 

Have fun!