Final Fantasy XIV's Island Sanctuary just got a bunch of new content for players to discover with the release of Patch 6.5. The newest area, called the Mother Lode, is host to four new materials islanders will need to gather to make progress. One of these is Island Yellow Copper Ore, which is used in various workshop crafts.
You won't need either of the new tools to mine this ore. You just need the Islekeep's Bronze Beakaxe, which you unlock fairly early at Sanctuary Rank 8. As soon as you unlock the new area, you should be able to mine the resource from Yellowish Rock nodes (X: 23.5, Y: 21.5). The cave they're found in is very small, so once you make it there, you won't need to hunt for them.

When it comes to the new island materials added in Patch 6.5, Yellow Copper Ore is one of the more straightforward ones to get. After you discover it for the first time, you'll be able to check its location anytime via the Gathering Log. This particular resource is used for the Isleworks Brass Serving Dish and Isleworks Gold Hairpin crafts, which you can set your mammets to work on at any time via the workshop.
It's likely that Island Yellow Copper Ore will be used in more crafts down the line, and possibly even larger-scale upgrades to the hideaway. For now, its use appears to be limited to a few things. Regardless, you'll probably be picking up a fair amount of it while gathering other materials in the area.