Not sure exactly what you mean about translating the name. There is no local equivalent of Elizabeth used as a name in Persian. The name of historical figures, like Queen Elizabeth of England, is rendered in Persian as الیزابت (elizābet). The Biblical figure of Elizabeth is rendered as الیصابات (elisābāt) or ایشاع (ishā') in Persian. These are very rare names that most Persian speakers have never heard of, so they might sound similar to giving a baby a rare Biblical name in English like Elisheva.
Please correct my mistakes in any language.