What is the importance of a very tangentially related minor in grad school applications? In an effort to keep this question generalizable, I'll try to be as vague as possible.

I'm majoring in a softer science (getting a BA), while doing research in a much harder science which could be considered a super-field to my major. This year, I did a year-long field project in my major field which counts toward a certain sociology minor that's at least somewhat related to my major. I'm interested in said minor and my planning said I could complete it, so I declared it. 

However, certain circumstances mean that I wasn't able to complete minor-satisfying courses in a timely manner. If I want to get the minor, I have to stick around an extra quarter. I'm not sure that expense is really worth it to me just for the minor.

I intend to apply to grad school in the harder field, a field in which I have a paper or two and a presentation at a major conference.

How important is an unrelated minor in applying to grad school? Unlike something like music or a language, there's probably not much I could do with it, even in my personal life. 



I read this question, but it's about minoring in a related field, not a distant field.



I've tagged this both United States and United Kingdom because I'm from the States but looking at a couple programs that would send me to the UK.
Except in unusual circumstances, a minor per se will matter for very little indeed in applying to graduate school.  There may be value in the particular things that you have done related to this minor (e.g., the field project) or reference letters you might be able to get due to your work towards it, irrespective of whether you actually complete the minor or not.

I would not however recommend delaying your graduation and grad school applications just to be able to formally complete your minor.