I do not think what we see in the show provides any evidence that there is any causal relationship between bending power and eye color, beyond tribal/familial/genetic ties.
We see many people who possess blue eyes, but not all are Waterbenders. Sokka is one prime example of this. In fact, the lack of reaction to Katara's blue eyes while she was journeying in the Fire Nation with Aang is actually evidence that blue eyes are not a mark of a Waterbender. She claimed to be a refuge from the colonies, and no one really thought anything of it.
In addition, in The Legend of Korra, we see several "mixed race" benders, and they do not support this "eye color causation" theory.
Tenzin has blue eyes, like his mother, Katara, but he is an Airbender.
Tenzin's children also display deviance from the "Airbenders have grey eyes" theory. Jinora, Tenzin's daughter, is an Airbender, yet she has brownish eyes.
Tenzin's newborn son, Rohan, has bright green eyes. He has not been shown Airbending as of yet, but Katara predicted that he would become the most powerful Airbender of the family.
I believe when it comes down to it, this is really just a confusion of correlation and causation.