The use of worry stones originated in Ancient Greece. They have an indentation and when you hold one between your index finger and thumb, rubbing it is believed to lessen your worries.
This action is a type of "stim" which can often create feelings of calmness and reduce stress levels. A stim (short for self-stimulation) is a repetitive body movement that self-stimulates one or more senses in a regulated manner. Rubbing a worry stone provides a "norming" effect and can be a very soothing behavior.
These "pocket tranquilizers" use a form of reflexology, or nerve pressure. Rubbing the smooth surface of the stone puts pressure on nerves within your thumb that may release endorphins, the brain's natural relaxants. You also benefit from the healing properties of the stone itself.