Eve was the mother of life because she brought death into the world

 Toras Moshe (Bereishis 3:20): Adam called his wife Eve because she was the mother of all life.Rashi explains that this goes back to the initial issue of giving names. Accordingly in agreement with our Sages (Avoda Zara 5a) : Come let us render gratitude to our ancestors because they brought death to the world. Because if they were still alive we in comparison to them would be like animals. But now that Eve brought death to the world she became the mother to all life because withot her we would be like we never came into the world.