Nearly all of something, the majority, as in The dentist was late; I waited for the best part of an hour. The adjective best here does not concern quality but quantity. [First half of 1500s] Also see better half, def. 1.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.