Jane's Genealogy Jigsaw
While I have included links to a couple of sites that explain the patronymic naming system, I want to add some personal observations. These are based on my own research in the church records and census records, so may only cover this very specific area (Horsbüll and Emmelsbüll parishes).
It is not all that difficult, at least not when refering to men.
Andreas Marcussen was the son of Marcus Bahnsen who was the son of Bahne Marcussen.
When they took permanent surnames, both Andreas and his brother Jens took their father's first name as a middle name and became Andreas Marcus Bahnsen and Jens Marcus Bahnsen.
It is when we talk about the women that it gets trickier. Especially in the earlier records, I have found that women did not really have surnames, but were identified by their relationships to the men in their lives. So, prior to marriage, a woman would be referred to as someone's daughter, often with her father's entire name written out, for example: Caren Carsten Hansens Tocher or Ancke Carsten Chrestensens Tochter.
In later records a woman might be referred to by either her father's given name with an -s attached, or by his surname, for example: Ingeborg, daughter of Carsten Momsen, is sometimes Ingeborg Carstens and sometimes Ingeborg Momsen.
Once a woman is married, she may be referred to by a possessive form of her husband's given name, for example, Caren, daughter of Redlef Frützen, and wife of Carsten Momsen, is listed as Caren Redlefs (in her marriage record), Caren Carstens (in her daughter's baptismal record), and Karen Carstens des weiland Redlef Frützen Tochter (in her death record).