Wondering what you might've meant when you said a more complex solution would be required for the eye to not leave the mesh. I understand that it would be important, but largely I'd like to know what 'key words' we can research to give us the desired result if we went ahead to fix a problem like this :)
Here's an advanced cartoon eye rigging tutorial from Chris McFall. It's older but the principles for a more robust eye setup still apply.
As for keywords, I googled "blender eye rig" and got a lot of results. Like Mark suggested.
Thanks Kent! The 'drivers' & 'deformation' were the exact wording I was looking for so that the eye doesn't clip through the geometry, even better to know there's a CGCookie tutorial for this, very helpful!! :)