Son of the seventh son is not a Christian doctrine. It comes from folklores. This refers to an unbroken line of only male descendants. The son of the seventh son is supposed to have mytical powers. He is often called the "cure".
The number seven has always been a very mystic or sacred number. It consists of four and three which are always viewed as lucky. Even among the Pythagoreans, these two numbers are accounted as lucky numbers. It is of great significance in Christianity. The number seven keeps recurring through popular mythology as well as daily life. For example there are seven deadly sins, seven heavens, seven Kings of Rome, seven seas, seven wonders of the ancient world and seven senses. Of all the numbers, seven is believed to be the most magical and mystical.
According to folk magic lore, seventh son of a seventh son is someone who is supposed to possess remarkable magical and healing powers. He is born with the gift of clairvoyance, with the ability to cast powerful spells on others. He is thought to be able to heal others by laying his hands on them.