First off, the "christian" that supposedly brought out the verse "an eye for an eye" should spend more time actually reading the Bible. If she actually read it, she would know that Jesus says: "When someone slaps you in the face, turn the other cheek" Jesus means that any wrong-doing that someone does to you, you should forgive them. (but not neccesarily withold punishment)
Also, The Bible tells to love our enemys and to forgive them-and basically that the old law (ie: the ten commandments) will not or ever get us into heaven. Only Jesus' grace and forgiveness of our sins can get us there. So doing "the right thing" doesn't get us in "good graces" with God.
So given that, why say "an eye for an eye"? If anything, the person should at least be allowed to live, even though its in a state of detention for the rest of his life. Who is to say, "you will die for what you've done!" That makes the jurors just as bad as the criminal himself. It says that in the end-times, Jesus will ultimately judge the world himself.
So now you know that I dislike the death penalty. Also, to add a point, I know of people who have recieved life-in-prison sentences, and through prison ministries recieved Jesus. They say that if it weren't for their life that they still had they would have never been truly happy. Hearing their story impacts me, and hearing that even though they were in prison for the rest of their life, but still happy, was astounding.
Life is precious, and no person should shorten it of his/her own free will. the only thing that determines how short your life should be is God, or yourself.