According to the text, Aunt Nicey was the one who delivered Doodle. In the matter of Doodle's survival prospects, Aunt Nicey's opinion differed from that of everyone else in the family.
She was the only one who didn't think that Doodle was going to die. Aunt Nicey believed that Doodle would live because he was born with a caul. Like many of her peers, Aunt Nicey believed that being born with a caul was a sign of good luck. A caul is a thin membrane that surrounds a baby's head at birth. Being born with a caul must be distinguished from being born en caul, where the baby is born still encased in the amniotic sac.
According to the story, Aunt Nicey also disagreed when everyone stopped calling Doodle by his original name, William Armstrong. She believed that caul babies should be given special respect because they might turn out to be saints.
So, Aunt Nicey's opinion differed in two respects from that of the rest of the family. Unlike everyone else, she believed that Doodle would survive his unpromising start in life. She also believed that Doodle should retain his original lofty name.