No matter where you are going to sleep the night, I'd definitely recommed taking a small midnight stroll outdoors - just to get an impression of the midsummer white nights.
In the morning, depending on your budget and how adventurous you feel, I could imagine a few options. For example, take a
commuter train, route I, but instead of driving it all the way down to Helsinki central station, stop at some smaller station. Like Vehkala or Pohjois-Haaga, which both have parks to walk around, rather close to the local train stop. (I'm not completely sure, but I'd guess the same train line is labelled P when it drives the same route but opposite direction, so you'd probably like to take that on your way back to the airport).
Or, if you'd like to see the actual Helsinki city, take a train or bus there. Near the central station you could either take a walk around Töölonlahti, or head west to Hietaniemi - area, which has parks (and a cemetary), a beach, and a few nice cafeterias here and there.
But, all of that naturally depends if you are suffering a jet-lag or something. So, back to basics; if you feel like skipping all the other activities, the only thing I recommend is a little walk outdoors either at midnight or in the small hours. After living here for more than forty years I still feel that there is something special in the light of the northern summer nights.