projection mapping is the best way to do an eye.

1/ set up the geometry (make sure the poles are facing outwards, in the direction you want the eye to look)

2/create your material, go to color- click the checkered map box, choose File AS PROJECTION and load up your image.

***ok projection mapping in hardware rendering doesn't always come out perfect, so you have to work with ipr render to see the results***

3/ in the workspace in the hypershade, locate the 3d projection node within the texture node itself, make sure it set as planar projection then bang the interactive placement button

4/ resize, scale and translate (rotate if you have to but you shouldnt need to if you've set up the geometry correctly) to have the projection facing forward infront of the geomtry. (see attached image)

(keep eye on poles!)

oh in answer to mhcannon's post- if its going to be animated, i would simply just set a reference point for the texture and the object rather than parenting it.- another option is just group everything into one.