One of the Gossip Stones Outside of Hyrule Castle say that "if you get close to a butterfly while holding a Deku Stick in your hands, something good will happen."
I've tried.. but ii must not be doing it right, but anyways what does happen??
Oh you're supposed to touch the tip of the deku stick to the butterfly. It's not very special. Just a fairy pops out and you can either capture it or have it heal you.