I also found that Elizabeth means "God is my vow," and that Lillian can be a shortening of Elizabeth. It would make sense, based on the story of Elizabeth in the New Testament (who gave birth to John the Baptist while she and her husband were at a very "advanced age", and then John the Baptist preached the coming of Jesus, etc....) that her name expresses a kind of commitment to God, since God allowed her, a barren woman, to give birth. It would be like saying, "God has answered my prayers" or "God is my promise." Hope that helps!
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