Actually, I think that both middle finger and little finger are normal name. They are both named based on how they look.
Index finger and ring finger are named based on how they are used (index basically meaning that the finger is used to point to things - and the index finger is sometimes also known as the pointer finger).
It is called the pinky. You don't need to say finger after pinky. The ring finger is called the ring finger because naturally, you put your wedding ring on that finger.