Djurgården is a nice place to walk around; there are a number of places where you can walk along the water. Gamla Stan is also a good place in the morning or evening. On Södermalm there are some terrific views over the city from the street called Fjällgatan. Or you could walk around Kungsträdgården (it is not that large) and stop for a coffee at Wayne's Coffee. If you like to swim, the outdoor public pool at Eriksdalsbadet opens early in the morning and it also has a sauna. It is near the Skanstull tunnelbana station. Östermalms Saluhallen which is a market opens around 9:30 in the morning and it also an interesting place to visit.
A number of the museums are open late one night per week in the summer.
Another evening suggestion is to go to dinner at one of the places where you can sit outside on the water. Take the boat to Fjäderholmen and eat at Fjäderholmarna's Krog or to Nacka Strand for Restaurang J. Both of these places are fairly pricey, but have really nice settings.