What area to stay in Athens

The whole area inside the walking route shown on the map below is well located and safe, even at night. Whether it's in May or at another time.

It's roughly around the Acropolis hill. The central part is mostly pedestrian

https://maps.app.goo.gl/hPBNLsk3GQa4PAyf6

The area on the map below (the little streets around Omonia Square) has quite a few cheap hotels, I think you won't like it, so avoid it. And apart from the National Archaeological Museum it's further away from the main sites and museums. Come back in a few years, it's undergoing gentrification.

https://maps.app.goo.gl/RibtzHCCVRAFJ2Cc9

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We prefer to stay in the Kolonaki neighborhood. Many good restaurants.

I really liked the Acropolis View Hotel when I stayed there last fall.

Thanks Jolie, Carol and Suki

There are several authentic Greek neighborhoods to consider other than the usual tourist spots.

Look into Thissio or Koukaki.

There are some hotels but a better option may be Airbnb where you can get apartments or studios at very good prices even better than some hotels.

Both are close enough to what most people want to see in Athens but quieter and where they'll be more locals than tourists.