Could you tell me what can cause a child to be born with only the right side of their heart?

Many factors: Start here Excellent resource http://www.medicinenet.com/congenital_heart_disease/article.htm

HLHS: The condition you are describing is officially called Hypoplastic Left Heart Syndrome. This defect occurs during first trimester the when the heart is developing. The cause of this defect is unknown and the only known risk factor is having a sibling with the same defect.

Mostly unknown: Hypoplastic left heart syndrome is most often a random occurrence.