SWMBO and I were visiting family graves at a nearby cemetary today (It's a Jewish tradition to pay such visits the week before the Jewish new year begins.)
The oval ceramic tile with her grandmother's photo on it had fallen out of its recess in her granite tombstone and fortunately I found it in the grass alongside the stone.
Whatever had been holding it in place for about 60 years finally gave up and let go. The remains of it in the recess are frangible and feel like some sort of "plaster".
The back surface of the oval ceramic tile is unglazed and the surface it will be attached to is smooth granite.
The "This to That" website isn't very helpfull, they don't mention "stone" and their references to ceramic refer to broken dishes and the like.
Thanks guys,
Jeff