During your time building up your Island Sanctuary in Final Fantasy XIV, you'll be needing a lot of materials to reach your goals. One of the materials you will need for several upgrades to your facilities, improved tools, and some handicrafts, is Island Mythril Ore. Knowing where to go in order to gather this material will be vital to progressing your piece of paradise.
How to get Island Mythril Ore in FFXIV's Island Sanctuary
Getting your hands on Island Mythril Ore requires you to enable the "Gathering" mode on your island, allowing you to click on Bluish Rock nodes in order to receive materials. You'll first have to unlock the Island Sanctuary, and increase its rank until you get the Islekeep's Steel Hammer to be able to extract Island Mythril Ore from these rocks.

Here are all the places on the island where you can find the Bluish Rock nodes:
Peak Point: This is the location your Sanctuary Gathering Log will point you to. Peak Point lays around X: 30.0 Y: 28.7. You'll find eight nodes in this area, which means you will need to hit 3 other nodes in order to make a full gathering route. There are plenty of nodes around here, so you won't have to go out of your way very far to respawn the Bluish Rock nodes.
Screengrab by Fanbyte via Square Enix Gentle Slope: Another good area to come to gather Island Mythril Ore lays just north of Gentle Slope, at X: 16.9 Y: 16.0. There are another eight nodes in this area. While it's not possible to make a mythril-only gathering circle here, there are a lot of nodes in the area that are also very useful for your sanctuary, making this spot the a very efficient one to gather at.
Screengrab by Fanbyte via Square Enix Rocky Reef: A less than optimal area for gathering Island Mythril Ore lays at the north of your island, at the Rocky Reef. There is five nodes spread along this area, making this the least optimal gathering spot to go to.
Screengrab by Fanbyte via Square Enix