I checked the Norwegian online censuses from 1801 to 1900 and “Lawson/Lawsen” does not appear. It is unlikely to be Scandinavian.
If your family came from England, then it probably is a corruption of “Larson/Larssen” (son of Lars) from the Scandinavian conquests. Many English -son names are like this. E.g., Patterson.
Without knowing what country, and even better what region of that country, your ancestors came from, it is all speculation.
Dunno. Best bet for some quick research: your local LDS Family History Center. Yes, the Mormons. They have, I am led to believe, some remarkable genealogical resources. You may also be able to find some of what you are looking for at the web site.