Generally speaking, it means "please tell her hello for me and that I wish her well." It's used for any situation, not just when someone is sick. It's similar to "send her my regards."

When someone is sick....if it is a female we would say "Give her my best" ....if it is a male who is sick we would say "Give him my best"....in this situation it means "Let them know that you are thinking of them and that you hope that they will get well soon."