None will be cheap, I think Oslo is probably the most extortionate but has the most spectacular surroundings. Copenhagen has a good selection of sights and a couple of theme parks (not as wild as Alton Towers etc. but the Tivoli is right in the centre) and even a beach. Helsinki is the least visibly exciting but if you get into it's history it becomes more interesting. You didn't mention Stockholm, which is a nice walkable city and also fantastically expensive. But Stockholm also opens up an interesting option. There are huge luxurious overnight ferries between Stockholm and Helsinki so you could have a day in Stockholm, night on ferry, day in Helsinki then another night on board back to Stockholm. I went across a few years ago and the cost of food (Scandinavian buffet) and accomodation on board was so much cheaper than on land that the saving covered the fare. You'll need to book in advance to get the best deal, turning up on the day may not even get you on at all, especially around weekends.

In my humble opinion do not go to Helsinki - so many better places. Spent an afternoon there and it was long enough. Extremely expensive as well.

Nivsy

I haven't been to Stockholm so can't comment, but both Copenhagen and Helsinki would make superb break destinations, though both are expensive (and always have been!)

Copenhagen

Tuborg brewery or Carlsberg brewery visits
Sweets museum
Tivoli Gardens especially if it's an evening with fireworks
Little Mermaid - but she is really VERY little compared to the photos
Day trip to Malmo in Sweden

Helsinki

The two Markets on the sea front
The T tram that does a tour of the city
Cathedral
The Church in the Rock - tremendous!
Zetor (make of tractors) bar
Visit to the islands - to see the old fortifications
Stockmann's department store

Do taste Aquavit in Denmark - caraway liqueur good for sorting out coughs - and Lakka in Helsinki ... a sweet liqueur made from Cloudberries. Lakka (cloudberries) look like raspberries but they are orange in colour and taste semisweet. You can buy Lakka jam at Ikea - very nice - and of course there are plenty of flavoured vodkas. You can eat reindeer, superb redcurrent jelly and the local equivalent of black puddings. Good buys in Finland are glassware - look for Aalto - ceramics, wooden toys and decorations and Hackmann cutlery. Beer is good but beware of Class 1 which is non-alcoholic and uninteresting. Stick to Class 3 which is like lager or Class 4 which is stronger. Lapin Kulta is a good brand.

Perhaps I would suggest you go to Finland if you manage to get good flights because it feels so much more 'foreign' than Denmark. In Finland English is widely understood and the Finns like the Brits ....... and their road signs are in Finnish and Swedish, because Finland used to belong to Sweden. They DON'T like the Russians who are their border buddies, but enjoy life to the full. There's lots of nightlife in Helsinki and although it can be expensive, I would seek advice from Lonely Planet or the Rough Guide. If the web site http://www.inyourpocket.com covers Helsinki, it's certainly worth a read. Have a great time, and what a fabulous present. You must be much loved!