Rockabye, or rock-a-bye, means to rock a cradle in order to lull a baby or child to sleep. When I say "rock", I mean to rock back and forth like a rocking-chair (not the thing that AC/DC does). :) Also, just fyi, you often hear (at least, in the UK), parents and children using the colloquial expression "to go bye-byes" meaning "to go to sleep".

I have only ever encountered the word in one usage, an old English nursery rhyme: Rock-a-bye bay on the tree top, When the wind blows the cradle will rock, When the bow breaks the cradle will fall, And down will come baby, cradle and all.

It's a real word, but I think it is just a kind of "baby talk." (I call it "baby talk" but I guess it is more correct to call it "child-directed speech.") In this case, it is used in connection with rocking a baby to sleep. It is an extension of the work "rock," in much the same as one adds some extra syllables to words to express affection to a young child. The "-bye" part does saying goodbye to someone whom (you hope!) is going to sleep. My mom sometimes wish me "beddie-bye," for example. Similarly "hush-a-bye" means "be quiet, stop crying, goodbye, go to sleep." In the spirit of adding rhyming or alliterative syllables, in a cooing tone of voice, instead of "bedtime," one might say "sleepy-time." Instead of calling a child "Billy" one might call him "My silly Billy" or "my Billykins."

It looks like an Arabic words "ركبى" means my knees !

It used to encourage a child or baby to sleep