There is more than just boring logs and vines to gather once you unlock the Island Sanctuary in Final Fantasy XIV. You need a wide variety of materials if you want to upgrade your sanctuary and reap its benefits. A solid gathering rotation can be very beneficial to operating your island at maximum efficiency.
Island Wood Opal is one of the prettier materials you'll want to add to this rotation. You'll need it for one of the many landmarks you can build, a handicraft, a tool upgrade, and a few Mammet-sized tools, making it an important material to the progression of your Island Sanctuary.
How to Get Island Wood Opal in FFXIV's Island Sanctuary
If you want to gather Island Wood Opal, you'll first need the Islekeep's Iron Hatchet. Without the right tool, all you'll be able to get from the Mahogany Tree nodes is Island Sap. Once you have the right tool upgrade, simply enable "Gathering" mode on your island and interact with Mahogany Tree nodes to gather Island Wood Opal.

Mahogany Tree nodes are plentiful, but most locations only have a scattered sprinkling of them. There is only one real farming location for these trees:
The Wilds: The best location to farm for Island Wood Opal is in The Wilds. (X: 20.0, Y 24.2). There are 11 Mahogany Trees in this area, so you can just gather them in one big loop without needing to hit any additional nodes.
Screengrab by Fanbyte via Square Enix