My experience is that many Americans have problems understanding the naming conventions used earlier in Norway, especially those among the peasantry and worker families. Usually you can not follow a family name in these cases. In the old days Norwegians were identified by their Christian name and their father's name: Hans Pedersen (or Pederson, Pedersøn), i.e. Hans, the son of Peder; or Siri Pedersdatter (or Pedersdotter, Pedersdtr.), Siri, the daughter of Peder.
In addition a third name (a farm name) was usually used. This last name does not necessarily identify a family or a relationship; it only tells where the person lived. You might call it an address. When a farmer is called Ole Olsen Li you can be quite sure he was farmer at the Li farm. If he moved from Li to the farm Dal, his name would change to Ole Olsen Dal. A farm labourer or servant was named in the same way, without being related to the farmer.
A tenant farmer (a cottager: Norw. husmann) was usually entered in the official registers under the name of the farm to which his home belonged, mostly with the Norwegian prepositions 'under' or 'ved' ('v/') before the name, or the suffix 'eie' (possession) after the name. Thus a cottager on the farm Lunde could be called 'Hans Nilsen Lunde' or 'Hans Nilsen ved Lunde (v/Lunde)' or 'Hans Nilsen Lunde-eie'. All these tenant farms (Norw: husmannsplasser) had their own names that were commonly known and used, but almost never in official registers. However, you may find exceptions here too. You could find something like 'Nerhagen v/Lunde' which means the tenant farm Nerhagen belonging to the Lunde farm.

Official registers sometimes use only the forename and the patronymic, sometimes the forename and the farm name, and sometimes all three names.

It was not unusual in Norway that a farm was split into two or more parts, each part being an independant farm. In this case it was common that each part was identified by a preposition indicating the part's location. Common prepositions in the Faaberg area are: Nordre (Northern), Søre (Southern), Midtre (Middle), Østre (Eastern), Vestre (Western), Øvre (Upper) and Nedre (Lower). These prepositions very often were placed in front of the farm name like: 'Søre Lunde' (Southern Lunde farm) or 'Nerhagen v/Søre Lunde' (tenant farm Nerhagen belonging to the Southern Lunde farm).

Through the years I have seen some examples of family histories where the genealogist has not been aware of these facts. Lots of effort has been put into a search for the wrong person. If one of your ancestor's name was Hans Pedersen he cannot be a son of Ole Pedersen (Pedersen is not a family name, but a patronymic), but he can be a son of Peder Olsen, or another man whose first name is Peder. I.e. you can rule out all persons with a first name other than Peder.

When a man and a woman married, the woman did not take the name of her husband as is common today. Both of them kept their forename and patronymic as before, and as the third name they used the name of the farm they were living on at any specific time.

There are of course some exceptions from this rule. Upper class families and many families in the cities usually used a family name. This family name could be the same as a patronymic, thus making it difficult to distinguish between a patronymic and a family name when we are dealing with families in the cities. You can also find some examples of farmer families that used a family name, but there are not many of them.

A complicating factor when we are talking about Norwegians that emigrated to America is what they called themselves after they had settled down there. Most of the emigrants changed their name somehow. Norwegian names are usually difficult to pronounce in English and a change was very often the only solution. The Christian name usually was somehow Americanized, while the last name could in some cases be changed completely. One example I have is a man from Fåberg with the last name of Bjørstadrønningen. In America he changed his name to Dalberg. It is understandable that he changed his name, but it is not easy to connect Dalberg to the Bjørstadrønningen farm.
Very often they took their patronymic as their American family name. But we have many examples also that they took their father's patronymic as a family name. This fact makes it even more important that there is enough information about emigrated Norwegians to be able to identify them.

Today all Norwegians use family names. Patronymics are not used any more. The transition started in the late 19th century, and in 1925 the use of family names was made compulsory by law. The last decades of the 19th century and the two first decades of the 20th century can be a difficult time to trace descendants because you do not know what last name a person used as an adult. Many family names today are originally patronymics that became family names. Practically all Norwegian family names today ending with -sen are originally patronymics.

As a conclusion, we can say that it is almost impossible to say anything about family relationships based on the last name when we go back to the 19th century and earlier. Grown children used different last names than their parents or brothers and sisters because they were living on different farms.