A non-derogatory matrix $A$ is one, whose minimal polynomial $m(z)$ equals its characteristic polynomial $p(z)$, where we apply the convention $p(z) = det(zI-A)$, while a matrix is derogatory, if they do not coincide. I have certainly never felt particularly offended when dealing with the identity matrix $I$, so seriously I wonder, where that strange name has its origin.
I would like to know about it, because I teach some advanced pupils at a german gymnasium school, who will be considering a field of study in the STEM-area, some linear algebra using examples from polynomial geometry. In that context, certain polynomials generate matrices with a double eigenvalue, which are either non-derogatory or diagonalizable, and where the non-derogatory ones are indeed preferable to the diagonalizable ones due to the particular application.
In order to make the rather involved notion more palatable to the audience, it is always good to have a story to tell about it, even if it turns out to be rather dull in the end.
My native language is german and I know of only one place in the german literature, where non-derogatory matrices get a special name. That is in E.Brieskorn "Lineare Algebra und analytische Geometrie II", where they are called "regular", which the author immediately distinguishes from another use of that word as invertible.
On the contrary, the english denomination sems to be well-established when refering to Google and I found it already mentioned in the first chapter of J.H. Wilkinson "The Algebraic Eigenvalue Problem" from 1965 without further comment, so it seems to have been established by then.