Gamla Stan and central Norrmalm put the Royal Palace, the Vasa and ABBA museums’ island of Djurgården, and Stockholm Central Station within easy reach, and suit a stay built around first-time sightseeing. Södermalm trades some of that centrality for a more residential, design-and-café-driven register with its own waterfront views. Östermalm sits quieter and more upscale, closer to the Royal Dramatic Theatre and the harbour where many archipelago boats depart.
None of the three is a wrong choice; naming the priority — landmarks and museums, neighbourhood character, or a quieter base near the archipelago boats — before booking tends to produce a more comfortable stay than assuming any one island is simply “Stockholm”.