First of all, the caul was the thin, filmy membrane or the remains of the amniotic sac, that covered Doodle when he was born. Aunty Nicey thinks it is important because she is very superstitious and believes in the idea that anyone born in a caul is good luck and is destined for greatness. This idea dates back to medieval times when this belief was widely held.
When Aunt Nicey delivers Doodle, he is born with a caul. A caul is a membrane that usually covers the head at birth, and in some religious traditional beliefs is known as "Jesus' nightgown".
Aunt Nicey feels that this is a sign, and that the family should treat Doodle carefully and respectfully, as he may someday be a saint. It is symbolic because Doodle does have a spiritual nature, and does seem to be pure of heart.