What would cause a small child to cough blood?

Bronchitis or Nose. Bronchitis is usually the most common cause of coughing up blood (hemoptysis). However, they may also have a small bleed at the back of their nose, which is dripping down their throat and subsequently being coughed up. The most common cause of nose bleeds in children is what we call "digital trauma, " a.K.A. Nose picking.
Hemoptysis. Throat irritation secondary to infection, bronchitis , TB infection, pulmonary abscess , pneumonias, embolism and rarely bronchiectasis , cancer or granulomas.