Final answer:
Brother renames William Armstrong "Doodle" because he is physically weak and frail, resembling a small, helpless insect called a doodlebug, and his endearing and peculiar ways affirm this nickname.
Step-by-step explanation:
The reason Brother renames William Armstrong "Doodle" is because he believes the name is more fitting for his younger brother. When William Armstrong is born, he seems so weak and unlikely to survive that Brother thinks of him as a doodlebug, which is the name of a small, helpless insect.
As Doodle grows, the name sticks not just because of his physical vulnerabilities but also due to his peculiar and endearing ways that are different from everyone else's, much like a doodlebug is different from other insects.