About the only three vocalic sounds for the standard Arabic pronunciation, you can check any reliable source for Arabic language. I have here the Gramática árabe, by Corriente (pp. 10-39), but the same information is ofered at Wikipedia:
As তন্ময় ভ said, when someone ask for *the* Arabic pronunciation (as it was the case of the OP), the pronunciation asked is the standard classical pronunciation (conserved in the Quran); specially when asking for "the correct pronunciation". Other dialectal pronunciations are possible, but in those cases it is necessary to put the gentilice before "Arabic": Moroccan Arabic, Lebanese Arabic, Egyptian Arabic, Syrian Arabic...
Leyla/Leila represents the Persian pronunciation ['lejla] (LAY-la, where LAY rhymes with "day" or "may"), adaptation from the Arabic Laila. This form of the name (and the Arabic original form) are known since the Middle Ages, known by Persian tales.
The Arabic pronunciation ['lajla] (LY-la, where LY rhymes with "my" o "by") was rendered mainly as Layla/Laila. In English, however, and due to the pronunciation of the group AY like [ej] (day, may), Layla/Laila is also usually read ['lejla]; that is why it is not unusual find that both spellings (with A and with E) are variants. In fact, they are cognates but not variants in the same origin language.
You can check Diccionario de nombres propios, by Roberto Faure, and manuals of European medieval litterature about the transmission of the Leila and Majnun story and One Thousand and One Nights tales in Europe.