Generally, Europe is an expensive place for a holiday. Also, if you are only going for 1 week, you aren’t going to see countries – your going to see 1 or 2 cities max. Don’t think that European countries are all small and you can zip between them in an hour. You can’t. Moving location costs money and time. If you change city/country too much then you won’t see Europe, you’ll just see the inside of a train station, airport or airplane for a week.
Remember, the first day of your trip is airport, long flight(s) (you’ll probably have a stop over in Dubai or similar), airport food, poor sleep, time change, jet lag, finding your hotel, going to sleep. Wasted day.
The last day of your holiday is similarly, checkout in the morning, fret about getting to the airport on time, waiting on trains/buses etc. then flying home. Wasted day 2.
Any change in your base city mid holiday will mean another wasted day on the travel.
Most cities in Europe have quite enough things to see and do to keep you entertained for a week. Most cities are also connected well to nearby cities that you can day trip to.
A day trip means that you go to the other city in the morning, spend the day there, then return to your base city in the evening. By doing this, you don’t have to move your heavy language and you don’t need to change hotels.
Another tip is that capital cities tend to be more expensive for hotels and food than other cities in a country. Going to the second or third city of a country and day tripping to the other cities is often a good choice to save money.
It might be a good idea to skip the city completely and head to a village instead. You’re going to miss out on good transportation for day tripping and miss the big museums and famous sights, but on the other hand you are going to be in the country side where walks in the forests and mountains are free. It will also allow you to experience the true local way of life, meeting real people and eating the real food, away from the globally identical shopping streets of most major cities.
Anyway, what location. Well Eastern Europe is typically cheap (Latvia, Lithuania, Bulgaria, Croatia, Romania, Czech, etc.) as is some southern places (Greece, Spain and Portugal). The north and west tends to be expensive (France, Germany, Switzerland, Belgium, Holland, Denmark, UK, Norway, Sweden) so steer clear of there.
short list for cheap European destinations would be:
Prague (Czech Republic)
If you would have to choose just one, it would be Prague. There is more than enough to do and to see in or nearby Prague.