Hmmm...I don't think I have heard of a flute performance minor, but you could minor in music. But really...do you have to minor in music? I know you really love music, but don't feel as if you have to show that by minoring in flute performance. I know how you feel. I really wanted to double major in piano performance (or flute performance) or minor in music, but I've realized that I could love and involve myself in music as much as possible without pursuing a degree in it. My suggestion is to find a college that will give you a lot of access to the music facilities even if you're NOT a major. Also, my warning: some colleges don't even give you that much access if you are a MINOR. Without a minor, you can still have lessons, take lots of music classes/theory/history etc etc, participate in the orchestra in colleges that give you lots of access...so think about it.
There are a lot of schools that offer the music minor. Usually you have to go through two levels (or so) of theory, two keyboarding credits, some PMI credits, etc. A lot of times, too, you don't even have to be a music major to participate in the ensembles.
I don't think it is possible to get a minor in "flute performance" if your MAJOR isn't another field in music. B.M. degrees (major and minor) in "flute performance" are generally restricted to specialized schools/colleges of music like those at the University of Michigan or the University of North Texas. They don't offer "minor" B.M. degrees to students in schools/colleges other than the School of Music
But at universities where you can get a B.A. (rather than a B.M.) in music, you can definitely minor in music. At one of the very best music departments in the U.S., many students minor in music while getting a major in another department (Biology) or college (Engineering). But it wouldn't be a minor in "flute performance." Instead, it would just be a minor in "music."
Most colleges have a general music minor that you can fulfill with ensemble and lesson credits, but getting into some of those classes can be very difficult. Lessons offered by flute professors are often very challenging, as they demand practically every minute of your free time to pratice. Most professors also require that you give a recital at the end of the semester, which can be stressful to prepare for.
I recommend that you either take private lessons with a flute grad. student or someone in the area and play in ensembles on campus. You don't have to pursue a minor in something to have credibility in the subject (in fact, most people are not impressed with minors).
Minoring in music is essentially equivalent to not minoring in it at all. Why? Because if you are doing music part-time and not as your main goal in life, then it is essentially something that you are doing for yourself(nothing wrong with that). As such, you should just get a good technical or other degree that you can use to live off of and do your music by yourself, part-time.(or maybe do something technical in music and do playing part-time)
I honestly never understood the whole reason for minors unless it was in something specific to your career(ie - music and religion might work if you planned to be a Cantor or something similar)
Some schools require a minor as a means of achieving their idea of a "well-rounded education." I agree with you, though: minors are a means of expressing your interests rather than your academic achievements.