"Eineinu" means "our eyes," not to be confused with the eyeglass franchise with a similar name. Though it is a simple word that follows some grammatical rules that are usual in Hebrew, it is a little confusing to people who don't speak Hebrew outside of prayer.
The singular word is "ayin." The Hebrew letter of the same name was originally a pictograph of an eye. The plural of ayin follows the rule of all body parts that come in pairs – no matter how many there are, they take the "doubling suffix" which is "-ayim." (Hands are "yadayim," ears are "oznayim," noses (because they have two nostrils) are "apayim.") Therefore, "eyes" are "einayim." When a suffix is added to this plural, "-ayim" becomes "-ei-." And the suffix meaning "our" is "nu," not to be confused with the Yiddish all-purpose question. Hence, "eineinu" means "our eyes."
(And, by the way, all body parts have feminine gender in Hebrew in spite of plural forms that appear masculine.)
The grammatical lesson aside, the inclusion of a specific bodily function of any kind in prayer is pretty unusual. In some of our more poetic prayers, physical activity is invoked metaphorically. The idiom meaning "efforts" (ma'asei yadayim) literally means "work of the hands." But the usage here literally means eyesight, as if to anticipate the common idiom "seeing is believing."
Eyes are the most fragile of our sensory organs and probably the most mysterious. We may understand more of the physiology of our eyes than ever before, but the reasons why we consider our eyes more credible than our other senses is still a function of their uniqueness. What we see is both internal and external simultaneously. Unlike our other senses, we cannot discriminate which stimulus we will allow to enter – we see everything we can and nothing we cannot. And modern technology not withstanding, mostly our eyes do not deceive us – Jacob could pretend to be Esau only because Isaac was nearly blind.
Our eyes are the portals of truth. To invoke them in prayer is to yearn for truth to be obvious.