Island Silver Ore is a type of rare crafting material in FFXIV. It’s currently exclusive to Island Sanctuary and can specifically be found during foraging expeditions performed after building a Granary. Thus, the material doesn’t appear in the Island Gathering Log and is mostly used in the production of Handicrafts.
You can “gather” an Island Silver Ore by first building a Granary. Then speak to the Excitable Explorer NPC (a mammet found in the center of the Home Sweet Hideaway zone). Finally, select “Mossy Mountains” from the list of available Expedition Area options. The fish will appear on the far-right side of the menu — under the Rare Resources square.
You can collect your expedition rewards from the Excitable Explorer after a specified time once per real-world day after putting in an order.