Patience is not considered one of the four cardinal virtues; sometimes expressed as justice, wisdom (prudence), courage (fortitude), and moderation (temperance). Although a wise person is patient and wisdom itself happens to be considered one of the four cardinal virtues, so maybe patience forms part of fortitude or self-control. Whatever the case, the Greek language appears to be the reason virtues are expressed as feminine.
"For there are four main virtues, wisdom, courage, temperance, justice. Each one of these [virtues] is a sovereign wielding authority, and the man that has acquired them [the virtues] is by the mere fact of doing so a ruling monarch, even if he be destitute of material resources." The four virtues are described with four terms of feminine gender. The passage continues with feminine gender as it refers back to these virtues:— Westminster Theological Journal 66/2 (2004), pp. 325-36.
Bible expositor Alexander MacLaren writes regarding James 1:4—But let patience have her perfect work, that ye may be perfect and entire, wanting nothing:
I might gather all that I have to say about this great queenly virtue of perseverance in the face of antagonisms into the one word of the Apostle, ‘I count them but dung that I may win Christ.’ ‘Forgetting the things that are behind, and reaching forth unto those that are before, I press toward the mark.’ ‘Let patience have her perfect work.’ Maclarens Expositions Of Holy Scripture
Jam 1:4. ἡ δὲ ὑπομονὴ ἔργον τέλειον ἐχέτω: “But let endurance have its perfect result”; the possibility of losing heart is contemplated, which would result in something being lacking; the words recall what is said in the Testaments of the Twelve Patriarchs, Joshua 2:7.
If patience were a masculine noun it would be written thus: ὑπομονος (hupomonos). However, ὑπομονὴ (hupomone) is a noun that takes the feminine form in Greek. Even though some would rather translate ὑπομονὴ as endurance instead of patience.
In English we would not say “her patience” if we meant “his patience”. However, we would say “let patience do her work” because this is considered a virtue and the Greek philosophers gave virtues a feminine form as per their language. The King James Version and a few other versions of the Bible have followed suit and used the feminine form, but most translations take a more neutral form of expression. For instance, the Modern English Version of James 1:4 reads:
But let patience perfect its work, that you may be perfect and complete, lacking nothing.
The feminine expression has no specific meaning. It is used similar to the male form “he” in a generic sense, when talking about concepts, rather than having to resort to the double use of the third person gender pronouns (“he or she”, “his and her”, etc) repetitively in long pieces of writing, when it is understood that “he” refers to everyone. Similarly, when it comes to virtues, the feminine form is often used.