Whit means a small tiny part of––as dregs is to liquid, whit it to everything else. You might feel the last whit of your confidence depart when your teacher hands out the 32-page final exam.
Don't confuse whit with wit, meaning mental sharpness or keen sense of humor. You might find someone described as a great wit to have not a whit of humor or understanding; not a whit of wit about them.