I don't believe Harper Lee ever explains the origin of Scout's nickname, nor does Scout's narrative reveal any clues. Born as Jean Louise Finch, father Atticus must have recognized early his daughter's innate sense of exploration and curiosity. The nickname "Scout" implies that she is someone who both observes and collects information, and she lives up to her name as a child when she joins Jem and Dill on their attempts to spy on Boo Radley; and as an adult narrator when she chronicles the events of the story. As for Boo's nickname, Arthur Radley Jr. is a "malevolent phantom" and a "ghoul": Never seen but undoubtedly alive (since no one has "seen him carried out yet"), he is like a ghost who only comes out in the dark of the night when he is less likely to be seen. Feared because of the rumors that abound about the carnage he creates on wild animals and pets, the ethereal nickname of "Boo" seems most appropriate.
Jean Louise Finch's nickname, Scout, implies that she is a curious tomboy. A scout is a soldier who goes on missions to gather information about the enemy. Similar to scouts in the Army, Scout also goes on missions to search for information. She sneaks into the spooky Radley yard to get a look at Boo and continually is getting herself into precarious situations. Scout is an appropriate nickname for a curious child who acts like a little boy throughout the novel. Arthur Radley's nickname, Boo, implies that he is a scary individual. As children, Jem, Scout, and Dill feared Arthur Radley because he was a reclusive individual who lived in a dilapidated home. The rumors surrounding Arthur made him seem like a grotesque monster which earned him the nickname Boo. Similar to a ghost, Boo scares the children even though he means them no harm. Arthur "Boo" Radley is similar to a friendly ghost throughout the novel.
Thank you so much! This really helped. I had the same ideas, I just couldn't really condense it the way you did. Thanks again!