As an undergrad I had a couple majors and a couple minors...no big deal. Ultimately all it meant was I took a lot of classes all over the place and went through the time and effort to declare them.
As a PhD student, though, declaring a minor is pretty uncommon. Hell, a lot of schools don't even offer the option. My understanding is that minors are usually for highlighting a certain amount of coursework in a closely related field to the one you're studying. Seemingly obvious pairings would be a PhD in Math with a minor in Statistics, or a PhD in Classics with a minor in Philosophy. However, unlike in undergrad (and really even at the Masters level), during a PhD the emphasis is primarily on research, not coursework. So why is a PhD minor a thing? Does it serve any purpose more than just another notch in your academic belt?
Personally, I am thinking about adding a Math or Statistics minor to my PhD in Computer Science, but I can't come up with a good reason other than "I'm interested in higher level math/stats and it's really applicable to the area of CS I work in." That's enough to satisfy me, I guess, but is there really any other reason to do this?
EDIT: I guess I should add that *in my case* I'm not thinking about a minor for minor's sake, but rather that I'm already taking a bunch of courses in that area so it just means an extra course or two. The question still stands though: why is it a thing? And, furthermore, does it carry different connotations in different areas (e.g. the humanities vs. applied sciences)?
Answer
In addition to the laudable goals of actually advancing one's education in a structured way, as several of the other posters have mentioned, it does serve to credentialize you in the minor department. Someone with a PhD in Math with a minor in Statistics potentially reads to me as very different than someone strictly doing a PhD in Math, even before I dig into their CV. That is one function - essentially keyword recognition of "I belong amongst both groups".
As someone who works in an interdisciplinary field, that can frequently helpful.
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