From where do we get the word "lullaby"?

"The English term lullaby is thought to come from 'lu, lu' or 'la la' sound made by mothers or nurses to calm chidren, and 'by by' or 'bye bye', either another lulling sound, or a term for good night.[2] Until the modern era lullabies were usually only recorded incidentally in written sources. The Roman nurses' lullaby 'Lalla, Lalla, Lalla, aut dormi, aut lacte' is recorded in a scholium on Persius and may be the oldest to survive.[3] It is also speculated that the term may come from "Lillith-bye." In theory, mothers would sing a song to their children at night to keep Lillith, an evil angel, away from their children."

https://en.wikipedia.org/wiki/Lullaby#Origins_of_the_term

From this, she becomes the demon responsible for the death of babies. In ancient times, one needed to protect against such demons; today, we blame other factors for the death of infants. To guard against Lilith, superstitious Jews would hang four amulets, one on the wall of each room of a newborn babe, with the inscription "Lilith - abi!" ["Lilith - begone!"] which some think is the origin, much later, of the English word "lullaby."

http://www.straightdope.com/columns/read/1513/whats-the-story-on-lilith-adams-first-wife

lullaby

c.1560, lulley by, from M.E. lollai, lullay, from lullen (see lull). Second element perhaps from by-by "good-by."
lull (v.)

c.1300, lullen "hush to sleep," probably imitative of lu-lu sound used to lull a child to sleep (cf. Swed. lulla "to hum a lullaby," Ger. lullen "to rock," Skt. lolati "moves to and fro," M.Du. lollen "to mutter"). The noun is attested from 1659.

http://www.etymonline.com/index.php?search=lullaby&searchmode=none