It sounds like a beautiful thing to do, but as far as I know, it’s not widespread enough to have a single answer. If you call the hospital’s Ministry or Guest Services, they might be able to tell you where their inspiration came from.
It *may *stem from old pilgrimage sites, where people would bring their sick relatives to pray for a miracle, or drink from holy spring water to be healed, or view a sacred relic, or touch the local stones and absorb sacred healing vibes from the holiness of the place. Of course when one went to a place like that, one would want a souvenir, so it became customary to take a small pebble or vial of water or stone or brick or something portable with you. These items would often be considered to have some sacred healing energy themselves, and if you brought one (or a dozen) back to your village with you, you could offer or sell them to people with ailments who couldn’t make the pilgrimage. To stop people from literally tearing apart the shrines, some places would bring in loose stones and scatter them around for the taking.