In the first two sentences, the subject has already been identified and therefore a generic pronoun is sufficient for further identification. This method is also useful in preventing repetition.
The sentence "He led her by the hand" sounds much better to our ears than "He led her by her hand". We know that is "he" is leading "her" so we have a visual image of what is taking place. The further information "by the hand" is actually technically unnecessary in sentence and thought completion but the phrase adds intimacy to the image. It is just a detail that adds to the reader's mental image of the scenery. In this sentence we see a man gently leading a woman to a special date, or a nice dinner. If the sentence would have been "He led her by the arm", we would have a different mental picture. Perhaps "He" was a policeman who had just arrested "her", or perhaps he was an abusive husband. The intonation changes with each example. The same is true for "I hit him in the jaw". We envision a fight between two men. If the body part were changed, such as "I hit him in the arm" we picture it more as a friendly, brotherly type of hitting interaction.
The third example you use is different because no subject has been identified. If you were to say "She closed the eyes" we would naturally assume that "she" was closing the eyes of another subject instead of her own eyes. The pronoun "her" adds identity to which eyes she closed and is necessary for sentence interpretation. Other sentences which follow this pattern might be "He rested his legs on the sofa." "He" wouldn't be resting someone else's legs on the sofa in common sense. "She bit her tongue after she spoke." "She" most likely would not be biting the tongue of another person after she spoke, meaning she wished she could take back what she had said.