Amy C.
Yelp
Listen, you can't really afford to be picky on an island where there are only two restaurants. Both restaurants, however, are rated quite well on Yelp. Truth be told, I had actually been leaning toward Dykarbaren for some arbitrary reason I can't remember until they told me that they would be closed on the Swedish National Day (6 June) and recommended Sandhamns Värdshus when I tried to make a reservation through email.
I highly recommend making a reservation for lunch. You're probably taking the 9:30 a.m. boat that leaves from Strändvagen in Stockholm that arrives at Sandhamn at 11:40 a.m. -- just in time for the lunch rush, so don't get stuck queueing behind everyone who was on your boat at this restaurant. We ate in their upstairs dining hall, which felt like a cozy bed & breakfast. Munching on the complimentary bread (god, what I would done to bring a carry-on full of Swedish bread back home...), I ordered the fried herring with mashed potatoes, lingonberries, and melted butter for 139 SEK or approximately $17 USD. I had already eaten the exact same dish in Stockholm (I'm assuming that this is a quintessential Swedish meal) and this tasted just as good, although be forewarned that there are quite a number of bones still left in the herring.
You can't go wrong with choosing Sandhamn as your Stockholm archipelago destination, so make it a perfect island getaway by eating at Sandhamns Värdshus for lunch. You'll need the energy to thoroughly explore everything Sandhamn has to offer before sundown.