I know of someone who pronounces this name "bur-NEES" (as if it was "Bernice"). In my frank opinion, coupling that pronunciation with the spelling "Berenice" gives the name a certain elegance; it just rolls off one's tongue that way. Pronounce it "ber-e-NIE-see" or "be-re-NEE-che"? No offence, but I'm afraid that doesn't hold with me.