淡菜 is only the name for dried mussels or dried sea mussels.
汉语大词典's entry for 淡菜 says:
贻贝的肉经烧煮曝洒而成的干制食品。味佳美, 以煮晒时不加盐, 故名。
So 淡 comes from the fact that 不加盐 (salt is not added).
菜 must be for the fact that it is a dish and not because it is a vegetable.