The mass of a body is a measure of the amount of content held within it and remains constant if no new mass is added to it or mass is removed from it. Thus, a person whose mass is 80 kg on the moon will still have a mass of 80 kg on Earth or anywhere else in the universe. Think about it this way. When the person is on the moon, he/she has two hands, two legs and all the other regular body parts. The same body configuration will also be present on Earth, without any change in the size or shape or their content. Thus, the mass remains same, unless growth or degeneration (or damage) to the body takes place.
What will change is the weight of the body. Our weight is the force with which a body attracts us and thus will depend on the acceleration due to gravity. The moon's acceleration due to gravity is 1/6 of that of Earth. Thus, a person weighs 6 times more on Earth than the moon. This person will have a weight of 80 kg x 9.8 m/s^2 = 784 N on Earth and 130.7 N on the moon.
Hope this helps.