GHAZZE, Lebanon — Amal Ismail El-Suleiman dropped a handful of creams and ointments on the floor of her home. They were just a few of the more than 15 she had tried on the skin disease that has covered her son Sariya’s face over the past four years.

The condition also manifests itself on the back of his neck and hands, and looks more like a burn — red and blistery — than a rash. It started four years ago after the family’s home, on the outskirts of Damascus, Syria’s capital, was hit by a rocket late at night.

In the chaos, Ms. Suleiman and her husband ran out of the house with their four older children and forgot Sariya, who was just a year old.