When she gave actual prophecies, Sybill Trelawney had a completely different "persona," not remembering what she said afterwards. Was she named after "the" Sybil (the one known for multiple personality disorder)?
JKR has stated this:
Sybill's first name is a homonym of 'Sibyl', which was a female clairvoyant in ancient times. My American editor wanted me to use 'Sibyl', but I preferred my version, because while it keeps the reference to the august clairvoyants of old, it is really no more than a variant the [sic] unfashionable female name 'Sybil'. Professor Trelawney, I felt, did not really qualify as a 'Sibyl'.
Going to the Wikipedia page for 'Sibyl', we see this:
The sibyls were women that the ancient Greeks believed were oracles. The earliest sibyls, according to legend, prophesied at holy sites. Their prophecies were influenced by divine inspiration from a deity; originally at Delphi and Pessinos, the deities were chthonic deities. In Late Antiquity, various writers attested to the existence of sibyls in Greece, Italy, the Levant, and Asia Minor.
So - no, she wasn't named after Sybil Dorsett; she was named after the sibyls of legend.