rock-a-bye

[to sway, move backward and forward] Middle English rokken "rock (a cradle), cause to sway back and forth; rock (someone) in a cradle," from late Old English roccian "move a child gently to and fro" in a cradle, which is related to Old Norse rykkja "to pull, tear, move," Swedish rycka "to pull, pluck," Middle Dutch rucken, Old High German rucchan, German rücken "to move jerkily."

The intransitive sense of "move or sway back and forth unstably" is from late 14c. For the popular music senses, see rock (v.2). Related: Rocked; rocking.

The earliest associations of the word were with slumber, rest, security. The sense of of "sway to and fro under some impact or stress" is from late 14c., especially of vessels in the waves (1510s); hence rock the boat in the figurative sense "stir up trouble" (1914). The sense of "swing to and fro in or as in a rocking chair" is by 1795.

"soothing song sung to infants," 1580s, noun use of the words lulley by (1560s), from Middle English lollai or lullay, a common burden in nursery songs, from lullen (see lull (v.)). Second element perhaps from by in good-bye or simply a meaningless extension.