Sounds like you've got it right, it'll be Hestan Isle, Dumfries and Galloway, and there was a copper mine there. Also elsewhere in the district at Enrick, Lauchentyre, Dromore, Drumruck, Tonderghie and Colvend. Information on the Hestan mines is taken here from "The Industrial Archaeology of Galloway", by Ian Donnachie, 271p, David & Charles, 1971:
"Hestan Island (NX8350): these old copper mines are located near the N end of the rocky island of Hestan, situated at the mouth of Auchencairn Bay. They were probably first worked in the late 18th century and re-opened by an English company before 1845. Ore was shipped to Swansea. On the W of the island two levels 15yd apart have been driven inwards from just above sea-level, and 30ft higher up there is another level. Further up, the sites of two old shafts can be traced and on the other side of the island there is further evidence of mining activity."
Hope this helps!
Chris