Wikipedia says (http://en.wikipedia.org/wiki/Dead_Sea) "Dead Sea" is the Arabic name (البحر الميت, Al Bahr al Mayyit); that'll be why it's used in English. The Jewish Encyclopedia says says (http://www.jewishencyclopedia.com/view.jsp?artid=175&letter=D&search=Dead%20Sea) the name "Dead Sea" is used by Pausanias, Justin, and the Church Fathers.

But apart from that, it's called Dead because it's so salty nothing can live in it. There's no fish, no seaweed, nothing; just salty water and dead rocks.