Today at the gym we collectively couldn't think of the name for the inside of the elbow and decided on "second armpit" 
Love it, going with that next time we do zercher squats
CROOK OF THE ARM
damn it, that's what it is....kind of a mouthful though
armpit secondus is better 
elbow pit
you really dont realize how little you directly USE that part of your arm until suddenly an entire group of adults is trying to describe how to hold something with your "second armpit"