Since the chest X-ray and blood tests are normal, serious structural or common infectious causes are unlikely. For a one-year-old, avoid over-the-counter cough medicines as they are not safe. Since your baby is one, you can give 1 teaspoon of honey to soothe the airways. Use a cool-mist humidifier and keep the home free of tobacco smoke, dust, and chemical irritants. Since the cough is persistent, I highly recommend consulting a pediatric pulmonologist or allergist. They can evaluate for underlying conditions like cough-variant asthma, environmental allergies, or gastroesophageal reflux, which often present with normal initial tests but require specialized treatment.