Does anyone have any record of names that were common for women in the 1300s BCE in the areas now known as Turkey and Italy?
Most of our knowledge of Anatolia in the 14th century BCE comes from Hittite archives, which I discussed here.
Die hethitischen Frauennamen: Katalog und Interpretation by Thomas Zehnder is an excellent compilation of female names from the Hittite period. He includes not only names from cuneiform texts but also names known only from hieroglyphic seal impressions. The first part of the book contains a morphological discussion of Anatolian names, and the catalogue of names begins on page 104. Each name is accompanied by a list of the texts in which that individual appears.
You should also take a look at the short but comprehensive Hurrian Personal Names in the Kingdom of Ḫatti.