Edithburgh Museum occupies the old McGlew and Adams produce store near the jetty, which is the right setting for a town built on shipping, salt, grain, gypsum, and fishing. The displays are local rather than grand, but they give useful texture to a place that can otherwise look like just another small seaside town.
Look for material on the port years, the salt industry, shipwrecks, schools, sport, and everyday life on the lower Yorke Peninsula. It is a volunteer-run museum, so check opening times before making a special trip.