Two main things:
-Perspective. A coach should be able to keep in mind that these are not just athletes, they're people. They're kids. They're learning, and gymnastics is (or at least should be) but one small part of their lives. Helping them to become stronger, happier, healthier people is of paramount importance; helping them to win meets is of negligible importance.
-Stage presence. Coaching, before all else, is theater. The first thing that a coach needs to do is engage their audience; only THEN can they actually teach them anything.
-(I guess after this comes all the unimportant stuff like technical knowledge, spotting ability, experience, blah blah blah. All this stuff is fascinating on an academic level, but trivial compared to the first two points)