The island of Riddarholmen should definitely not be left out on your visit in Stockholm. Especially not if you like some architecture and nice views. Noone lives on this island anymore, it's only the famous Riddarholmskyrkan and some buildings with offices, a big part of it belongs to the gouvernment and the Supreme Court. It is really worth taking a walk there. If only to enjoy the best view on Stadshuset or also because there are not less than six palaces around Birger Jarls torg. If you like architecture, take a stroll there and also visit the courtyards that you can find when walking through some iron doors.
And while there, visit Gamla stan, only a bridge away and find out what else to discover in Stockholm.