An Island Hammerhead is a rare crafting material in FFXIV. It’s currently exclusive to Island Sanctuary and, more specifically, foraging expeditions performed by 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 instead “gather” an Island Hammerhead 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 “Bending Beaches” 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.