I can't really speak to the practical benefits of minoring in linguistics. (Perhaps the Linguistic Society of America can; see http://www.linguisticsociety.org/what-linguistics for some of their articles about why people should major in linguistics.) I can say that I took a course in linguistics, fell in love with it, and decided to major in it, but that isn't going to be everyone's path.
What the benefits are probably depends on what you plan to teach, and where. What you will probably gain from studying linguistics, from the point of view of working in education, is a general understanding of how languages work. This may help you understand why students make certain errors in their writing, no matter what you're teaching. But it'll be particularly useful if you're teaching students whose first language isn't English, or if you're interested in teaching another language. It can even be helpful if your students speak a non-standard dialect of English. So if you decide to teach calculus to middle-class students who sound like newscasters and whose ancestors came over on the Mayflower, none of these things will apply, and you might not gain much from having studied linguistics. But chances are that you'll find it useful at some point.
I'm also a firm believer in the idea that what you study in college, especially when it comes to minors or other not-for-your-major courses, isn't necessarily about its immediate practical uses, but rather about learning to think about new things in new ways. My courses on Russian science fiction and the history of Indian mathematics (two courses I really took!) aren't "useful" to anything I do, but they had the benefits of making me look at things I wouldn't have thought to look at, from perspectives I don't usually have. If you don't like science courses (and I often didn't), linguistics can be a brand new way of thinking about a familiar subject--language--in scientific ways. If you've always taken language for granted, linguistics can be a way to get yourself to really think about it, examine it, and appreciate it in ways you never would have otherwise. These things are useful no matter what you do, but they're certainly useful for someone going into education, which is often all about finding ways to interest students and get them to think about things from a new point of view.
I know this might not be completely helpful--someone in education might be able to give you a much more practical answer. But hopefully it'll offer some perspective!
A minor in Linguistics will provide you with an in depth understanding of the workings of languages. It will also provide you with critical thinking and organization skills that are beneficial in many jobs. Many of the courses you take in the linguistics field will help with creative problem solving skills and understanding the inner workings of human communication. It can also be very beneficial when working with foreign speakers as you will better understand the mistakes they make when speaking a foreign language.