My father has a blog with a genealogy theme,
Genealogy's Star. He posts about notable sources, new records, updates in New Family Search, legal issues involved with genealogy, and other related topics. In a post yesterday mentioning the variety of name misspellings in New Family Search, he said:
It might help to know that Ove Christian Oveson changed his name to Overson when he came to the U.S. from Denmark.
One of his readers saw that and replied:
If Ove Christian Oveson was Danish, then I'll eat my nose if his original name was indeed Oveson. No way. Either -sen or in older times -søn, not -son.
Well, okay. But "JP" didn't stop there. He or she continued:
For all my interest in the Overson line, I don't recall if I have made any but the most cursory look into the Danish records, and I have not been planning on returning to the Tanner (Overson) line on this blog until 2012. Here is the comment I wrote after my father sent me a link to JP's comment.
Wow. That's so cool, JP. I've done Swedish research on the other side of my family, but I don't know if I've ever gotten around to looking into the Danish records. Thanks for sending those! I'll have to put up a blog post tomorrow or Friday with those two records and a link to your comment.
As far as the -son and -sen problem: it was not unheard of for Danes to change their name to -son when they came to America. One notable example was Mormon Church Historian Andrew Jenson. He was Danish through and through, but used the -son spelling his whole life. I've assumed the immigrants changed to -son because it was the more American spelling and they were interested in assimilating and having their children be as American as possible.
Here is a post with Jens Andreas Ovesen's death notice:
http://theancestorfiles.blogspot.com/2010/08/jens-oveson-obituary.html
The death notice uses the spelling "Oveson" and "Oversen." His gravestone says "Ovesen."
(Make up your minds, people!! :)
The descendants of his sons Lars Peter and Ove Christian use the spellings Oveson and Overson, although as you note in your comment, it should probably be Jensen.
Thanks again for the info!
Amy (James's daughter)
http://theancestorfiles.blogspot.com/
http://oveovesondiary.blogspot.com/

Here is a summary of the information in this post:
1. Ove Oveson was born as Ove Christian Jensen, before the Danes switched to a non-patronymic name form. At some point, he began using his father's patronymic as a surname, with the Americanized -son ending instead of an -sen ending. An "r" was also added in the middle of the name. He used that until his brother Lars Peter convinced him that it would be more respectful to their father to use the "Oveson" spelling. (Why not go all the way back to "Ovesen" as a show of respect to their Danish heritage?) Some of Ove's descendants go by "Overson" and some by "Oveson." I believe that Lars Peter's descendants go by "Oveson."
2. Here is a summary of patronymics that my father linked in a subsequent post:
Sørensen, John Kousgård. Patronymics in Denmark and England. The Dorothea Coke memorial lecture in Northern studies, 1982. London: Published for the College by the Viking Society for Northern Research, 1983. (PDF)
3. JP pointed out
Statens Arkivers Arkivalieronline. This is a site containing Danish parish registers and censuses. The State Archive has certain minor restrictions on the use of the records, but it is a wonderful site.
4. Here is Ove's birth record. I have not had the time to sit down and decipher it yet, besides making sure that it is the correct person.
5. Here is his father Jens's birth record. I have not translated it either. (I'm going to have to pull out my Scandinavian research and handwriting guides. I'm a little rusty!)