Here are some thoughts on that: https://www.al-islam.org/ask/why-is-wearing-a-ring-stone-beneficial-to-m...
I am not sure if there is any specific mention of amethyst or pyrite in the textual sources.
There are a couple narrations in praise of sapphire, for instance, that it dispels poverty, and that Imam 'Ali (A) used to possess one.