BEIRUT—Back home in Syria, Omar al-Kurdi was always the good kid. He got top grades. He helped his disabled father manage a small shop selling cigarettes and snacks. He hung laundry for his mother.

Now, as a refugee in Lebanon, Omar is again the good kid, this time as the family's breadwinner, working 14 hours a day in a vegetable shop. His salary of $15 a day pays for rent and food.

"At...