Excerpt from Asher Intrater:
When Jacob married Leah, his third son was Levy. His name came from Leah’s saying that her husband would “closely accompany” her – Genesis 29:34. This is the root of Levy לוי. Leah meant this in a soulish way, but God – in His sovereign election of this family – used the name to achieve a spiritual destiny. Those who became the priestly tribe, took the name Levy as one who “clings to,” “joins to,” “closely adheres to,” or “accompanies” the Lord.