Got it! The term for a family’s linguistic idiosyncrasies is oikolect:
Don’t you love it when you find the Right Term(tm)?
The reason I’m excited is that I found it in a neat way. Went to Babelfish to find the Greek prefix for family. Though I can’t read Greek, the prefix looked like “oiko” so I did a search for “oikolect” (using the logic behind “idiolect“) and it just so happen that the word has in fact been used in exactly the sense I was thinking about.
Man are those Internets great!
Will eventually need to look into published research about oikolect. Come to think of it, the term might have been mentioned in one of my course in linguistic anthropology. So I should probably feel bad about not remembering it. But one of my most recent resolutions is to not feel too bad about my limits. And it’s working!