It's an idiom that basically means "getting attention." "Did that strike a chord with him?" "Yeah, it really got his attention!" In the context above, "minor" means "a little," so "striking a minor chord" means "getting a bit of attention."

If something "strikes a chord", it is well received. "The song struck a chord with the audience", "that business plan really struck a chord".

Being a man who's holding a sunburst Strat on the profile photo, I'm pretty sure I know what you were thinking when it comes to minor chords. I haven't heard it used in a musical context, so I would say, unlike the two answers above, that it has more to do with when you know something or something sounds familiar or it matches your sensibilites, although it's certainly well received. And no, you can't strike a diminished chord! Well, YOU can but not in an expression :)