It's all about how you approach the creation of a character, Prop or scene element, before you begin animating. Unfortunately, if you have started animating with a Character, and you have used grouped objects. then you are lost. You will physically have to go through every sequence where a new frame exists of the Eyes, and update them.
Its the importance of using and creating Symbols and Movie clips. When you first create a character and you are happy with each element, such as the eyes, the mouth the head, hands and so on, create a symbol, either graphic or Movie clip, of each element, depending on how detailed you want to be. This way in future, you can make an update and it will carry through every instance of that symbol in your movie.
If you have created the animation, surely you have Symbol instances? You should be able to make a change that carries through.
Or is your animation created entirely with grouped objects, or are you animating classically using shapes? Fair play if you are, but unfortunately you must update every frame instance of the eye image.
I have not done anything yet... Im just anticipating the problem. So the lesson is that you should always make symbols of the pieces in an animation, right?