Philosopher’s stone? No.

I'm totally thinking of the beryllium spheres from Galaxy Quest

A ring? No.
Sci-fi? No.
Valuable? Yes is probably the best answer, but I’ll warn you that it may be misleading.

Unique? Yes.

Currently existing? Yes.

Fixed location? Yes.
Monument? No.

In a museum? No.
Mythology? No.

As I'm feeling generous, here are a few hints.
1. You'll probably need to come at this from both angles. That is, you need to identify what kind of thing the MINERAL thing is, and you will also need to figure out the ABSTRACT connection.
2. Most, probably all, of you won't actually know the specific wording of the AOTC. But an equivalent identifying phrase is acceptable (and, once you've figured that out, the AOTC would be a quick google away).
3. Despite (2), most, probably all, of you know the ABSTRACT connection and most—though probably not all— have in some sense encountered the MINERAL AOTC.

Statue? No.

In human body? No. (Unique, remember)

Made of stone? Yes (though that’s perhaps an odd way to put it, given you’ve established that it’s not human-made).
Religious? No.

Cairn? No.

Smaller than double-decker bus? No.
On Earth? Yes.
Geographical feature? Yes.

> 1 country? No.
Cultural significance? Yes.

Uluru? No. For free, I'll tell you that the cultural significance is probably not going to be particularly helpful to you.