Green’s Dictionary of Slang

bug n.1

[SE bugaboo]

1. [mid-16C+] a person, esp. one who puts on airs; thus bug’s words, boasting language.

2. [1920s+] (US) an insignificant person; an irritating person.

3. see boog n.

In phrases

big bug (n.)

1. [19C] (orig. US) an important person, an aristocrat, esp. one who considers themselves to be one and acts accordingly.

2. attrib. use of sense 1.