We had the same problem on the mountains on our home island. It seems there needs to be Stone and/or Ice Golems along with obsidian for the mountain biome to have detectable silver. At lease that was our experience.

What I do now, is after detecting the silver, I dig down and expose the vein and mark it on my map with an icon & name (silver), but I do not tap the vein with my pickaxe. A single hit with a pickaxe will stop the wishbone's pulse detection sound. I do this when I am not prepared to mine and transport the silver ore, so when I go back I can find the ore easily enough by using my map, in case I forgot to bring my wishbone. I don't tap the vein, so it will still ping if my son is looking for silver in the same area. We don't always play at the same times, and our map sharing feature does not always work as it should.