Mental toughness can be taught. I think what you're talking about is a kid that is calm under fire. When she's trapped in the corner she has the presence of mind to make a good decision, hit the open man, or save the possession by calling a timeout.
Kids that are mentally tough simply do not panic! Today I see coaches going crazy on their players yelling and throwing things. That makes for an environment of panic and kids respond by being spastic, overly wound, and make many mistakes because it feels to them the gym is falling in on them. Positive energy, a vision and belief that you will be successful from the head coach will go a long way to making a kid be mentally tough. I love to hear a coach yell at a kid and tell them they missed a layup.......I'm pretty sure they know they missed it no use beating them in the head with it. Fix fundamentals in practice, manage players emotions and composure during a game.
The mentally tough miss a layup and go on to their next job, they don't slap the floor, blame the refs, or a teammate. They simply go on to the next job or task at hand. To me mental toughness, confidence, and composure go hand in hand. Each one brings about the other.
I really think it can be taught!