Her Islamic name is Marjana.
The story of the story 'Ali Baba is an interesting one. A Maronite Christian woman from Aleppo (Syria) named Hannah Diab told the story to Antoine Galland, who wrote it down in French. Much later, a manuscript in pseudo-grammatical Arabic was acquired by the Bodleian in 1860. It turned out to be a modified translation of Galland, its author Jean Varsi, a Frenchman attached to the French mission in Egypt. Burton based his translation on the fake Arabic version, and exaggerated the "picturesque turns and novel expressions of the original in all their outlandishness," creating a 'pseudo-archaic style' that is alien both to the style of the Arabic 'original' and to any recognizable style in English literature (See "The Arabian Nights II," translations by Husain Haddawy, 1995, W. W. Norton & Co.).
Is the story Arabic- or Maronite Christian- or Persian- or? This is digression, but it is a question that affects Sindbad and 'Ala al-Din (Aladdin) as well.

"Tin Pan Ali" seems to be a review with several mis-used or bowdlerized Islamic names, including 'Margiana' and 'Fatimona'.

Also see Antoine Galland, "AliBaba et Les Quarante Voleurs."
Antoine Galland, "Les mille et une nuits." Contes arabes.
Antoine Galland, "Histoire d'Alladin ou la Lampe Merveilleuse."
Many reprints, the originals published in 1694 and ff.