We have been to Copenhagen, Finland and Sweden and are going to Sweden and Norway this summer. Ryanair does very cheap flights to both Stockholm and Oslo around £50 per person return plus luggage. Finland was more expensive to fly to and we used FinnAir.
Accomodation is expensive in cities but reasonable out of it, we are spending around £100 a night mostly for 4 of us this summer renting houses / apartments but not in big cities. Used AirBNB and Booking.com. Vegetarian food was excellent in Finland, think there will be good options everywhere - only issue you get is kids menus don't always have vegi option. What does he want to do? Cities, fjords, wildlife, action sports. Of the capital cities Copenhagen and Stockholm would be my preference, for a country Norway has the widest range of things I have found and most interesting scenery.
If he likes white water rafting, canyoning, rock climbing, cycling etc then in Norway around Voss is good with Bergen worth visiting, also we are doing Oppdal, then Ulsteinvik / Alesund then Oslo and there we have moose, musk ox and reindeer safari, canyoning, wilderness camp with bow and arrows, ax throwing, rock climbing, zipline, fjords, seals, otters and odd whale, kayaking / motorboats. In Finland we did brown bears in wild, have that in Sweden too. In cities for us it would be more £200 a night or stay 5 miles out at £100.