Reading around the subject of Lillith, the alledged, rebel first wife of Adam, I stumbled upon this link and many others that link the custom of singing babies to sleep as an ancient Jewish superstition.
Some of the research could probably be argued but looking at it from the WT perspective and mirroring their usual reasoning, I could imagine the following...
... it appears that the word "lullaby" originates from the the hebrew "Lilith-abi", which means "Lilith be gone". Jewish mothers used to sing these songs to their babies in bed in hopes that they would be protected from this baby snatching demon called Lilith. Most likely the superstition originated as a way to protect infants for what we know today "crib death". As the legend goes, three angels were to protect children from Lilith. Jewish women habitually hanged three amulets on the walls near the baby's bed with the images of these angels. It would make sense to relate this amulets with modern day toys like crib mobiles and others that are usually hanged on top of cribs.
Faithfull followers of Jehovah should consider carefully whether their customs are in tune with bible teachings.