Parshat Chayei Sarah begins with the passing away of Sarah, followed by the arrangements for her burial and the burial itself. The first pasuk starts with “And Sarah lived” and then immediately goes on to state in the second pasuk “And Sarah died.” Isn’t it strange that the parsha is named “Chayei Sara - The Life of Sarah” - and yet the parsha opens with her death? One would think that the parsha would, like a eulogy, start from the beginning of her life and end with her death. Furthermore, why is the parsha called “Chayei Sarah” – “The Life of Sarah” and not “The Death of Sarah”? (The idea that the first word of a parsha is necessarily the most significant word and therefore becomes the title is wrong. In fact, Noach begins with “ אֵלֶּה תּוֹלְדֹת נֹחַ” and yet it is not named with those words. And there are many such examples.)
The Lubavitcher Rebbe explained that the name of a parsha is not chosen by chance, rather it is a one word description of the primary concepts and themes discussed in the parsha. Hence, “Chayei Sarah” is focused on one goal, which is that Yitzchak reach spiritual greatness. This was Sarah’s main dream, so it is only appropriate that the parsha named “Chayei Sarah” discuss the life of Yitchak, who was the “realization of Sarah’s spiritual vision.” By raising Yitchak in the manner Sarah did and by teaching him her principals Sarah lived on vicariously through Yitzchak, the apotheosis of Sarah’s life. When Sarah died she only died in a physical sense, for she remained extant through her son Yitzchak, making this parsha in truth, the life of Sarah.