Sibylla was a prophetess (the original sibyl) who lived at Marpessus near Troy and was devoted to serving Apollo, who inspired her to make her predictions. She gave her name to all the prophetesses called sibyls, so that the meaning of her name will be the meaning of this word. Unfortunately, the derivation is not easy. Plutarch proposed an origin in Dios, 'of Zeus' and boule, 'counsel', so that the name means 'counsel of Zeus'.
Saint Jerome saw the name as originating in theou boule, 'divine counsel', with this being an Attic Greek expression for which the Doric would have been Siobolla. Eric Partridge favours a link with Latin sibilus, so that her name may mean 'the hisser' (presumably mouthing predictions).
In any event, her name first appears in the writings of Heraclitus, the Greek philosopher who lived in the sixth and fifth centuries BC. The name was steadily popular until quite recently (often with the spelling Sybil) as a Christian name. Its 'boost' in the nineteenth century may have owed something to Disraeli's novel Sybil published in 1845.