With only 5 days you need to choose carefully and budget your time wisely. I recommend the "mandatory" visit to the Parthenon along with the new Acropolis Museum, as well as a visit to the amazing National Archaeological Museum. Have a meal (or at least drinks) at the top of Lycabettus Hill at sunset. Ride the funicular to the summit. Take a daytrip excursion to the nearby island of Hydra, which is still charmingly traditional because no motorized vehicles are allowed. Ferries are frequent every day. Do some serious shopping in the Plaka.

The Parthenon and the Acropolis Museum are on the to-do list; "drink at the top of Lycabettus Hill at sunset" and a "daytrip to Hydra" is exactly what I was looking for! Thank you!

You'll find the funicular near Kolonaki Square. Enjoy!

1. Benaki Museum

2. Ilias Lalaounis Jewellery Museum

3. Hydra

4. Shopping in Kolonaki and sitting in a cafe in Kolonaki Square drinking wine wearing Prada sunglasses and throwing my Jackie O attitude.

Thin

I guess as you mention near-by islands, you do not refer strictly on Athens or Athens city center.
In addition to recommendations other members provided, I would consider some exploration of the sea side suburbs/cities, for example half beach day at Voula followed by lunch/coffee and posh shopping in Glyfada or maybe heading to Palaeo Faliro and Piraeus areas. Many diverse harbors and marinas with different ambiance each. Plus maybe for example you can combine strolling around Marina Zeas in Piraeus with a visit to the very interesting near by Piraeus Archeological Museum. Piraeus has some really good shopping too, brand names and expensive shops or bargain shopping depending on the specific area.
On a different note, I do not know your daughter's age and if this might be of interest, but on Spata near-ish the airport there is a very cool Attica Zoo Park worths considering too.
Near/around the city center, I like to take my time and walk around, and explore hidden gems of cafeterias etc, for example some people insist that Monastiraki is mostly about tacky shops, but ask around for 6 Dogs bar complex and check where the "underground" garden is... Suddenly you'll be found on an open-air cosy yard where you can enjoy a beverage and people watch... A different aspect of the most touristy spots' ambiance, only 100 meters away from the popular square. Ermou street starting on Syndagma Square is maybe the most famous shopping street in Athens. If you are much into shopping you may want to head on Shopping Centers/Malls that are to be found mostly on Northern Suburbs though eg Maroussi, OAKA area etc...
In terms of ruins outside the very center, there are many of them, but most are complicated to reach. I guess Temple of Posidon on Sounio is the most popular half day trip. Nice one but lots of time spent on transit, so would not be on the top of my list. And no, I do not think a day trip to Delphi is a good idea as some people suggest! I would try to cover sites at the very city center on this trip!
Of course, as someone already pointed out, you have no time to see and do it all, you have to make some choices. Note that Acropolis site ticket its self, covers 5-6 additional sites and museums! Do not underestimate time needed to check "basics" in leisure. A good strategy is to head on the sites as early morning as possible in order to avoid crowds and heat, then visit air conditioned museums or have a lunch break around hottest part of the day, and why not...have a refreshing siesta before you continue your exploration on afternoon. Experience will be much different than rushing around from place to place and try to fit it all.

If you like history, I would take a day trip to the Peloponnesian Peninsula. You can fairly easily visit Mycenae, Tiryns, Nafplion, Epidaurus and Ancient Corinth in a day. I hired a driver on the internet prior to my trip. He picked me up at my hotel at 8. I had already established where I wanted to go and what the price would be. He had an air conditioned Mercedes sedan and was very nice and spoke perfect English. He took me where I wanted to go and waited for however long I wanted to stay. He wasn't a guide, but I didn't really want a guide as I had done extensive research and knew what I wanted to see. If you are shoppers, I might cut out Tiryns and spend more time in Nafplion. It is a beautiful town right on the water with lots of shops. I spent several hours there and had lunch. Even with all of that, we were back in Athens by 6pm.

Thank you for all of the fabulous suggestions!

Go to Syndagma Square and watch the protests.

We took Key tours to Mysenae/Nafplion/Eppidaurus one day and Delphi another day. We had fun, saw a lot and the tour operator is top notch.

1. HydraHydraHydra. Did I say Hydra?? Leonard Cohen's old hangout. Spend a day there and you'll feel like you escaped Athens for much longer. It's about a two hour ride on the hydrofoil from Piraeus.

2. The National Gardens. A 40-acre public park within walking distance of the Acropolis. It's a great place for a picnic or a glass or two of chilled white wine.

3. The neighborhood I got lost in last year, and I don't know what it's called, but it's in back of the Acropolis as you walk toward Syntagma Square. It seemed a little bit like the Athens version of Soho- lots of small shops with handmade goods, art galleries, tiny cafes. Before you reach the Plaka.

4. The antique market at Avissinias Square, even if it's a bit tacky. I loved talking to the vendors, and looking through postcard bins of Athens and the islands. There's lots of stuff to sort through ( clothing, beads, records, silver) if you have the patience and interest.

5. A drink at dusk, either at the restaurant in The Grande Bretagne, or The Athens Gate, both of which offer unparalleled views of The Acropolis lit up at night.

6.Dinner at ITHAKI on the Vouliagmeni peninsula. Amazing seafood,great crowd, gorgeous views of the water, where Jackie might have hung out with Aristotle. And I'm only half-kidding. One of the best meals we've had in Greece. However, a FYI: It's about a 20-30 minute taxi ride from the center of Athens, depending on traffic.

1. Another very nice garden, that has been recently completely redone and is even bigger than the National Garden is Pedion tou Areos near to the National Archaeological Musuem, and opposite the Radisson Blu Park Hotel

2. Keramikos Area, a very popular area with great restaurants, cafes and life music , and above all, not touristy at all....
An ideal place for being among Athenians.
Just take the metro ( blue line) and get off at Keramikos station.

3. Kifisia suburb + the Mall, shopping centre
take the metro ( green line) to Kifisia.
Get off at Nerantziotisa and do some shopping at the attached huge Mall, called "the Mall", there are lots of restaurants and cafes there as well..
Then continue with the metro to Kifisia, which is a wonderful, green posh suburb.
Walk around and discover the wonderful old houses of wealthy Athenians that date back to the beginning of the 20th century, have a lunch in one of the fabulous restaurants and cafes, and do some shopping.. Kifisia has great shops of all kinds!!

4. take the bus to Nea Makri, a small village on the sea, in the north of Athens ( about 45 min away from Athens)
A great place to be on the sea, with many restaurants, cafes etc and not touristy !!! There is a beach too, even though you would need a car to reach the nicest beaches of this area.

5. Dinning at Mikrolimano in Piraeus, if you like fish, there are some excellent Fish Restaurants there, this is a small port with yachts. Do not ask the reception of your hotel for restaurant recommendations, they will send you most probably to all the wrong places.

Athens has a plethora of great restaurants.. Athenians rarely eat in the touristy restaurants of Plaka...

Weadles has given you great advice , if you like old things and antiquities try the market at Avissinias Square on a Sunday. This is the busiest day, when Athenians visit the market combined with a frape afterwards....
You will see frape on every cafe table.. it's our favourite drink, especially in the summer, cold instant coffee, with milk and sugar ( or without) shacked with ice cubes... we now have also newer versions.. like cold Cappuccino or cold Espresso... ( you can order them as Freddo Espresso, or Freddo Cappuccino, both taste great, and can only be found in Greece)

Forgot to add :

There is now a special action in the historical centre of Athens that involves restaurants and cafes.
In order to attract Athenians, the city of Athens has arranged super low prices in the following areas of Psirri, Monastiraki, Thisseion ,Plaka and Exarchia.
This action takes place every Tuesday, and you can find menus for 10 Euro, a drink at the price of 5 Euro or a coffee at the price of 2 Euro... ( when it's regular price is 4-5 Euro)

Clausar,

Thank you for the information about the menu pricing. Perfect timing for me. I will be there next week. Any recommendations for wonderful restaurants/cafes near the Acropolis? I don't mean to hijack the post, but...

Weadles,

Here are some of my favourite restaurants in the Acropolis/Makrygianni area

Trapezaria : new restaurant with very good new greek cuisine.
my number 1 choice, when in this area

www.trapezaria.gr

Attikos- Mina : good greek cuisine and amazing view from the
terrace.

Garibaldi 7 - near Acropolis

Aglio, Olio, Peperoncino: good italian restaurant next to the
metro station Acropolis
Porinou 13 - Makrygianni
Tel 2109211801


Melilotos - Monastiraki : greek traditional cuisine - like
homecooked food.
Kalamiotou 19
Tel : 2103222458 absolutely central location

http://www.melilotos.gr/ ( only in Greek unfortunately)
They have very long opening hours....


I hope this is enough.. unless you want more....

Would you be interested in cultural events as well, should there be something of interest?


Have a wonderful trip and enjoy Athens and Greece to the outmost. !!

Nikolas

clausar/Nikolas,

Thank you so much! Yes, I'd love to hear about cultural events taking place during the week of June 4th, although some of the time we'll be attending Posidonia.

And many thanks for the good wishes. We always have a wonderful time in Greece. Best wishes to you for the election... and after.