Whit means a small tiny part of something. If you don't even have a whit of courage, you are most likely a chicken.
Whit indicates a bit of something, and usually something abstract like a feeling or emotion. 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. Let's hope you have more than a whit of wit in your personality!