Shakespeare got the name from Samuel Harsnett‘s Declaration of Egregious Popish Impostures (1603), where one reads of 40 fiends, which Jesuits cast out and among which was Fliberdigibbet, described as one of “foure deuils of the round, or Morrice, whom Sara in her fits, tuned together, in measure and sweet cadence.”
Shakespeare uses the name in King Lear. In colloquial speech, it’s come to mean someone whimsical or ‘flighty’ – someone who flits from one thing to another, scatterbrained and garrulous – which also means that it’s associated with someone gossipy. It’s derived from late Middle English.
One wonders, of course, why any word develops in any particular period – were people particularly gossipy in the period between 1450 – 1475 (at the time of the origin of this expression)? Did it have anything to do with gibbets, or gallows, places of execution where criminals were publicly hung (although gibbet can refer to any instrument of public execution) and, therefore, became a source for gossip. Early forms of gibbeting involved suspending the prisoner in a cage hung in a public place, where they would die of thirst, and where, again, they would be displayed as a public spectacle. There might also be a marine origin, having to do with a gibbet sail, which flapped in the wind, and therefore an association made between flapping sails and flapping tongues.
Now, I think, we tend to favour the less sinister implications of the origin of the term flibbertygibbet and simply use it as a descriptive term for charming addlepates – which you can look up yourself.