The shining in the eye trope, as you call it, has been around for decades; I even remember seeing it in some black-and-white anime. It is a way for the animators and mangaka's to show the character's intent, and it has many spin-offs like the glowing glasses, which usually means the wearer of the glasses has a malicious intent.
It has a lot to do with the common saying of a gleam in your eye. I am sure you have been told this or something like it at least once in your life. Japan has a similar saying, quite possibly adopted from watching US movies and cartoons.
Sorry, the explanation isn't as interesting as "why the Japanese think people are talking behind their back when they sneeze."