Relaxing isn't the only thing you will need to do on your Island Sanctuary in Final Fantasy XIV. If you want to not only unlock your island, but also increase your island rank and rake in large amounts of gil, you will need to gather plenty of materials in order to keep up with demand, and keep your facilities upgraded.
Island Copper Ore is used for a wide variety of things. You will need this material to make tools for your mammets, upgrade your own gathering tools, keep your facilities up-to-date, and it's also used in a wide variety of Island Handicrafts. This makes it a very important material to keep in stock, so it's important to know where you can get your hands on a steady supply.
How to get Island Copper Ore in FFXIV's Island Sanctuary
Your first step will be to enable "Gathering" mode while you are on your island, so you will be able to interact with any node and get materials from it. The node you are looking for is the Bluish Rock, which can be found in several locations. In order to extract Island Copper Ore from these nodes, you will need the Islekeep's Stone Hammer.

You can find Bluish Rock nodes in the following locations:
Peak Point: If you look at your Sanctuary Gathering Log and click the Island Copper Ore icon, this is the location you will be pointed at. Around X: 30.0 Y: 28.7 on the map are eight nodes containing the sought after mineral. You'll need to hit three other nodes in order to respawn the Bluish Rock, but there are plenty of things to gather on these hills, so you won't have to go far.
Screengrab by Fanbyte via Square Enix Gentle Slope: Island Copper Ore can also be found in the group of eight Bluish Rocks that lay around X: 16.9 Y: 16.0, just north of the Gentle Slope mark on your map. This being another group of eight, you will once again need to gather an additional three nodes in order to respawn the first Bluish Rock you hit, but this area has plenty of useful nodes to pick from, making this spot very efficient.
Screengrab by Fanbyte via Square Enix Rocky Reef: Your last option is all the way in the north of your island, at Rocky Reef. This area has five Bluish Rock nodes, and they're spread quite far apart, making this the least optimal location to get Island Copper Ore.
Screengrab by Fanbyte via Square Enix