I'm pretty sure what you've described are biologic gems called "Operculum" also known as "Shell Cat's Eye". They have a flat white calcite base that has a spiral pattern and on the reverse is the dark operculum center spot - formed from some univalve snail-like molluscs - (the technical name of the mollusc is the "Turbo Petholatus") to protect their body parts against predation. The Operculum dark centre consists of a convexed dome of dark green with yellow / brown external eye-like pattern.
The polish surface displays reasonable good luster and was used for jewelry purposes in the late 19th and early 20th century.
Debbie, here is an early 20th century Operculum bracelet Click Here If you google around "Operculum" there's quite a selection.
If it was Operculum jewelry that your Mom's Dad made all those years ago (and the dates and description do seem to fit) - what a fantastic gift for your Mom!
