Hoping the Danish wizards can help me out once more.
The Møller branch of the family has consistently used Møller as a surname, not patronymics. My initial assumption was that if I went far enough back, I'd find an actual miller and that would be the source of the surname.
Instead, I've stalled on Thomas Lauridsen Møller, whose occupation was consumtionsbetjent (customs officer, as I understand it). What's especially confusing is that the family uses patronymics in the 1787 census (Lauridsen, Thomasen, Thomasdatter), and the baptism records that I've been able to find for Thomas' children list him as "Lauridsen".
It looks like they switched to using Møller between 1794 and 1798. Thomas' daughter Kirsten is listed as Thomasdr in her confirmation (1794) and his other daughter Mariane is listed as Thomasdatter Møller in hers (1798): https://www.sa.dk/ao-soegesider/en/billedviser?epid=17123888#164242,27580491
Thomas' burial record uses Møller: https://www.sa.dk/ao-soegesider/da/billedviser?epid=17123888#164242,27580444
Now, I suppose I could be conflating two different families -- the Lauridsen/Thomasen family and the Møller family -- but all of the names, ages, and places line up. So why the change? Was Thomas' father Laurids Møller and the names were just recorded inconsistently? (Entirely plausible, except I can't back it up -- I don't have anything for this family before 1779, Kirsten's birth.)
Any insights or new information would be greatly appreciated.