Well, because someone, most likely her parents, named her Nene (duh!).

Jokes aside:

It’s a pretty standard medieval/early modern period girl’s name, which were constructed by combining two kana characters. These names can be classified as childhood names (called warawana or yōmyō). Unlike the proper names of adults, although they may be written in kanji, there are no specific kanji associated with these syllables.

Moreover, unlike boys, who received a proper adult name during their coming-of-age rite, girls usually retained their name for life. (Japan’s premodern naming practices are so incredibly gendered...)

The major exception has traditionally been ladies of very high rank—third court rank and above—who, in court society, qualified to establish their own household administration (mandokoro) (1). These received a name ending on -ko, such as Yoshiko. The name Yoshiko is recorded in the proceedings of the occasion during which she also was elevated to Junior First court rank in 1588, where she is referred to as Toyotomi no Yoshiko; but, considering that her promotion to Junior Third rank occurred three years earlier, one may also suspect the name was already given to her at that occasion (2).
Otherwise, O-ne (Ne+honorific prefix) is most likely a nickname or call name (azana) of sorts, Ne an abbreviation used in signatures.

I don’t know what to make out of Nei, or Neiko, since I have no references telling me about the nature of the sources in which context these appear in comparison to the others, and how they chronologically correlate to usage of the other names.

(1) Hence, she became known as Kita no Mandokoro.

(2) Or they just had to improvise the name on the spot. The usage of the Toyotomi name here I find unusual, and Yoshi is quite clearly taken from her spouse.