This is likely to be difficult to pin down with any certainty due to the difficulty in providing anything other than approximate dates. However, the logical place to start looking is with those civilizations which first developed writing, i.e. Sumer and Egypt (Indus and Vinca might also be candidates if we were able to decipher their scripts, but so far we haven't).
Neith-hotep is a likely candidate as she was probably either the mother or the spouse of one of the earliest known recorded male names, Hor-Aha (circa.31st century BC), 2nd pharaoh of the First Dynasty. Her name is taken from the goddess Neith of the cosmos (and much more) plus hotep, meaning "to be satisfied, at peace". This Livius article from 2017 states:
It is believed that Neithhotep outlived her husband Narmer and
temporarily presided over Egypt until her son Hor-Aha came of age.
Another theory is that Neithhotep was married to Hor-Aha and became
regent to their son Djer [3rd pharaoh]. This assumption is based on an interregnum,
mentioned on the Stone of Palermo, between the rules of Hor-Aha and
Djer. This interregnum lasted one year, one month and fifteen days and
may indicate a short-lived regency.
Toby Wilkinson, in Early Dynastic Egypt, favours her as the mother of Hor-Aha but this possible one generation difference is unlikely to make any difference in attempting to determine the first named female as the next oldest, named, Egyptian females are probably Herneith (disputed by some) and Penebui, consorts of the 3rd pharaoh Djer, and thus a generation or two later than Neith-hotep. Next, there is Merneith, wife of the 4th pharaoh (First Dynasty) Djet. However, there are other names (such as Khenthap) which could be considered for the period immediately after Neithhotep; our evidence is simply too limited to be certain.
"Jar sealing impressed with name of Queen Neithhotep ca. 3100 B.C.". Source: The Met Museum
Probably the 'go-to'academic reference for women in early Egypt is Joyce Tyldesley's 'Daughters of Isis: Women of Ancient Egypt'.
There doesn't seem to be any real 'competition' from Sumer, either. The oldest named Sumerian female appears to be Puabi (possibly a queen or priestess) from circa. 2600 BC.
