It means a frivolous, flighty, or excessively talkative person. It’s a fine word to throw out, in the appropriate circumstances, though there’s a risk of tripping over all those syllables. That’s no doubt why it has had so many spellings.
The original seems to have been recorded about 1450 as fleper-gebet, which may have been just an imitation of the sound of meaningless speech (babble and yadda-yadda-yadda have similar origins). It started out to mean a gossip or chattering person, but quickly seems to have taken on the idea of a flighty or frivolous woman. A century later it had become respectable enough for Bishop Latimer to use it in a sermon before King Edward VI, though he wrote it as flybbergybe.
The modern spelling is due to Shakespeare, who borrowed it from one of the 40 fiends listed in a book by Samuel Harsnet in 1603. In King Lear Edgar uses it for a demon or imp: “This is the foul fiend Flibbertigibbet. .. He gives the web and the pin, squints the eye, and makes the harelip; mildews the white wheat, and hurts the poor creature of earth”.
There has been yet a third sense, taken from a character of Sir Walter Scott’s in Kenilworth, for a mischievous and flighty small child. But despite Shakespeare and Scott, the most usual sense is still the original one.