In the SCA, as in the world at large, an increasing number of people are interested in "gender neutral" names, that is, in names that could be used by men or women. So far, I have found onlly a few such names in Arabic. Thus, this article is very much a work in progress.
So far, I've found gender neutral names only in two works. One is Consorts of the Caliphs, written in the 13th century CE by ibn al-Sāʻī and translated by The Editors of the Library of Arabic Literature and edited by Shawkat M. Toorawa (published by The Library of Arabic Literature in 2017). This source gives three names, Bunūn, Faḍl, and Isḥāq.
The other is The Book of Strangers: Medieval Arabic Graffiti on the Theme of Nostalgia written in the 10century CE by Abu ‘l Faraj al-Isfahani and translated by Patricia Crone and Shmuel Moreh (published by Markus Wiener Publishers in 2000), and Islamic and Middle Eastern Geographers and Travellers vol.1 ed. by Ian Richard Netton (published by Routledge in 2008), which show Laila/Layla/Laylā/etc. as a male name.
Starting with Bunūn, we find both a male and a female Bunān in Consorts of the Caliphs, note that section 14 is about the female Bunūn, while the male Bunān is mentioned in passim in a few places.
The Faḍl in Consorts of the Caliphs, section 13, is clearly a female, but as shown in https://dragonstrike.neocities.org/names/battuta/ismbat.html#kunyas and dragonstrike.neocities.org/names/ibnmunqidh/ismmun.html, males were sometimes given this name.
As for Isḥāq, the woman with this name is the subject of section 12, while its use as a male name can be found in https://dragonstrike.neocities.org/names/ibnalbanna/ismsalb.html and dragonstrike.neocities.org/names/battuta/ismbat.html.
One thing to note about these three names; the women are all concubine slaves. While it looks like only women in such a situation had traditionally "male" names, there is no guarantee such gender neutral names were restricted to such women.
Laylā/Lailā/Layla/etc., however, is in the opposite situation: usually it's a female name, but I have found some examples of it as a male name. In The Book of Strangers section 58 starts with, "Abū 'l-Ḥasan ʼAlī ibn al-Kalwādhī known as Layla..." In section 74 he is called simply, "Abū 'l-Ḥasan Layla..." In section 75, there's a man named Laylā ibn Mūsā Fayādha.
In Islamic and Middle Eastern Geographers and Travellers vol.1, on page 190 we find "...and went swiftly to Nīsābūr, where Lailā ibn-Nuʻmān had just been killed..." (NB: this is reprinted from "Byzantina Metabyzantina 1(2) (1949): 7-37", in an article named "Notes on the Risala of Ibn-Fadlan" by Robert P. Blake and Richard N. Frye.)
For Layla as a female name, I point to (inter alia) the article in The Encyclopaedia of Islam, 2nd edition, on Laylā al-Akhyaliyya, a Arab poetess of the 7th century CE.
Again, let me say that this is very much a work in progress; I hope to add more examples of gender neutral Arabic names in the future.