If seeing the midnight sun as part of a crowd isn't exactly your thing, don't panic! You can also go to North Cape in March or April, when it's not too cold and when this particular "ends of the earth" is all yours, or nearly.
When I went there with two friends, not only were there only about a dozen of us there to enjoy the landscape, but we were also let off the entrance fee, which is normally obligatory (at that time anyway) for everyone.
In my humble opinion, it would be a pity not to go to North Cape if you are exploring Lapland, whether you are in a group or on your own. Even if you don't spend the whole day there, the countryside you see while getting there is magnificent and the high cliffs and endless horizon of the place are breathtaking.