The name does not appear in the Bible. It was probably a diminutive of “Elizabeth” (Exodus 6:23; Luke 1:5) which came from the Greek name Ελισαβετ (Elisabet) which was derived from the Hebrew name אֱלִישֶׁבַע ('Elisheva'), meaning “God is a vow”. In the Old Testament, Elisheba was the name of Aaron’s wife. In the New Testament, Elizabeth was a relative of Mary and the mother of John the Baptist.
“Lillian” is now more commonly believed to be related to the name “Lily”.