Hereditary Lines and Inheritance
While the entire world does not share the same customs, because of the dominance of the Hyperborean Empire and their culture, much of the known world shares a similar method of determining familial names and inheritance.
Inheritance and Familial Succession
Inheritance of Nobility in the land most commonly goes in order of firstborn. A noble might disown their firstborn, giving the titles of inheritance to the next in line, but that is uncommon and often weakens the families standing. While the firstborn will be next in line to become the family's patriarch/matriarch, others are often given prominent positions of power to maintain the families strength. Because of this method of transferring power, those who gain the mantle of family leader often reach out to their relatives in an attempt to maintain the cohesion of the family. Sometimes distant family becomes stronger than those who bare the name, leading to infighting or even those who change their name in order to make a new name for themselves.
Children of lesser members of the family are sometimes given important positions of power within the family, or are often married to other families to form alliances. Those married off would generally get lesser positions and take the last names of the family they marry into (regardless of gender). Peasants and those that do not marry for nobility purposes do not have a set method of deciding their names but often mimic that of important families and nobility. Some regions keep the last name of the male heir, others combine their last names, both parents keep their original and alternate children's names, some do not have familial names, or simply decide on some other metric.