Hi

For Athens, the Attalos Hotel - http://www.attalos.gr - is showing an early booking rate of €80 for a double with a balcony. Book now to get this rate - ask for a room with an Acropolis view. You could get lucky! The hotel has a wonderful rooftop bar and is a few metres to the Monastiraki metro stop.

Athens :
A for Athens, top prime location directly on Monastiraki Square, 20 meters away from the metro station where the blue line meets the green line.
The blue line connects Monastiraki with the Airport, the green line will take you to Piraeus...
Hotel is brand new, selling at a rate of 115 Euro right now

Sweet Home Hotel, located directly in the Plaka and an excellent hotel, selling right now at the rate 100 Euros,

The Athens Gate : has an excellent location next to Olympieion, National Garden the Acropolis and the New Acropolis Museum. Closest metro is the red line ( either Syntagma Square or Acropolis )

The club scene of Athens is concentrated in Kerameikos (same named metro station - blue line) or Gazi ( a short walk from Monastiraki. Psirri and Thisseion with many restaurants and cafes are both next to the first 2 hotels that i recommended.

From Syntagma Square you can also take the tram, to the coast line of Athens. Glyfada is a popular suburb with excellent night life. In the summer many clubs are opening in this area.

******************************************************

Mykonos

The clubs and the nightlife are located in Mykonos town... Most hotels are bit outside of the town...
Here are some recommendations :

Ilio Maris, Alkyon, Harmony, Portobello, Pelican.

They are all within your budget

Some beaches can be reached with the public bus ( Agios Stefanos, PLaty Gyalos, Psarrou) while to get to the most popular ones ( Paranga, Paradise, Super Paradise) you will need a bus-boat combination ( bus to Platy Gyalos and then boat)

Also consider http://www.athenscypria.com in Athens, which is just off Syntagma Square. Their luxury and superior rooms have very large bathrooms (by Greek hotel standards). I've stayed there 3 or 4 times. It has a good breakfast buffet and free wi-fi in the lobby.

I can't offer any hotel suggestions for Mykonos, as I've only taken rooms when getting off the ferry, but have found reasonable ones for under €50/night.

You can get a bus to Paradise Beach from the station up near the windmills (Platia Yialos?). I've also taken a bus to Agios Ioannis Beach from there. These buses run frequently during June. I understand you can get buses to Evia Beach from the station near the harbour, but I've never tried it.

Buses are leaving from 3 points :

1. next to old Harbour to Agios Stefanos, New Harbour
2. Agios Loukas Square ( Fabrica) to Ornos, Agios Ioannis, Paranga, Paradise, Platy Gyalos, Psarrou
3. From the square behind the OTE building ( don't remember the name...) to Ano Mera and Elia Beach.

Have a look at http://www.ktelmykonos.gr/index-en.html

If you plan to visit both Islands, look the Volcano-View hotel because they a loyalty with a big group of hotels in Mykonos (Saint John, Palladium, Nissaki) etc and you get a discount.

If you visit first Mykonos, then you get the discount for the Santorini's hotels (Volcano View, Petit Palace) and so on..

Or have a look here
http://www.mykonosantorini.com/

In Athens, Philippos, & Acropolis Select, both on the south side of the Acropolis area quiet,convenient to sights and metro. I would never stay at the Attalos. Much too noisy there.

OP asked for an area in Athens, that is convenient after clubbing.....
Best area is Monastiraki and Thisseio/Psirri, with most hotels being located in Monastiraki. ( and within walking distance to the clubs)

Philippos and Acropolis Select are nice, but not conveniently located.

We stayed at the Hotel Amazon in Athens. Very convenient fro transport to and from the airport as it's only a couple of blocks from the metro at Syntagma and and easy stroll to the Plaka and Monastiraki the other way. Don't skimp on Athens... take a day off Mykonos and add it to Athens.

In Athen's we really loved the location of Athen's Gate. And, the view of the Acropolis from the terrace is spectacular!

In Mykonos, we stayed at Hotel Leto for it's location and waterfront view. It has a nice pool when you want to cool off and is easy walking to just about anywhere in town. I liked not having to walk uphill after dinner or after exploring town.

We sent our son to both of these on his honeymoon and they too enjoyed both for these reasons.