- Eve: As in Adam and Eve. She is mentioned three times, where one of the three times comes from a direct quote written by Nephi. The other two references are direct quotes from Lehi (as recorded by Nephi).
- Mary: As in Mary, the mother of Jesus. There are two references to her, and both references appear in direct quotes that Mormon included in the Book of Mormon. One is from king Benjamin, who is directly quoting the words of an angel. The other is from Alma the younger , and it too appears to be a quote from an angel/the spirit.
- Sariah: Nephi's mother. All references to her are from the writings of Nephi, which Mormon included unabridged.
- Isabel: A harlot who tempted Corianton, the son of Alma. The only reference to her is from a direct quote from what appears to be something written by Alma to his son Corianton, and Mormon included it without abridgment.
- Abish: The subject of this post.
What is interesting when we look at this list is that four of the five women who are named explicitly in the Book of Mormon are named in passages where Mormon quotes them in their entirety meaning he did not change, shorten or abridge them. This means that of the five only Abish's name is included explicitly by Mormon. Even though Mormon mentions some individual women, and some incredibly faithful and interesting women, he does not name them. Even with Abish, after he gives her name, he refers to her as "the woman servant" instead of by name.
Let us look at the one verse where Abish is named to see if we can figure out why Mormon chose to include her name.
16 And it came to pass that they did call on the name of the Lord, in their might, even until they had all fallen to the earth, save it were one of the Lamanitish women, whose name was Abish, she having been converted unto the Lord for many years, on account of a remarkable vision of her father— (Alma 19:16)The verse refers to her as a "Lamanitish" woman, a term used only one other time in the Book of Mormon in reference to some servants who were servants to same king served by Abish. It is not clear why they are called "Lamanitish" and not "Lamanites". Some commentators have suggested that this distinction may indicate that the servants were a different ethnic group than the Lamanites, but there is too little information to make a definitive statement one way or another. But this still does not answer the question, why did Mormon include Abish's name when he told this story, since he did not include any other female names.
To see if we can sort this out let us turn to the Bible. The name Abish is similar to Abishag, a woman who is mentioned in the Bible in association with King David. There is also a male version of the name, Abishai, who was King David's nephew. In the transliteration into English all three share a similar part, namely "abish". Perhaps if we look at the Hebrew meanings of Abishag and Abishai we might figure out something about Abish.
The meaning of Abishag (אֲבִישַׁג) is unclear with some sources giving the meaning as "the [Divine] Father (?)", while others render it as "father of error" or "the father wanders". Abishai (אֲבִישַׁי) is similar with various interpretations, but its meaning is more settled than Abishag. Different sources give the meaning as "father of a gift" or "father of gifts", with apparently one source rendering it as "my father is Jesse" (for reasons discussed at this link). The only common thing in the various proposed meanings here is the "father" part ("ab"- אֲבִ).
So at the moment the best we can get for the meaning of Abish is "the father [?]". But this is encouraging considering the context. When Mormon mentions Abish he says that she was converted to the Lord "on account of a remarkable vision of her father." So apparently there was an important story involving her father, which Mormon may have known about, but did not include and thus the name Abish had special meaning given the interpretation of the name, the story of Abish's father and the situation that Mormon was writing about.
Let us see if we can take this a step further. In Hebrew "ish" (אּישׁ) is the word for man (as in "Then the man said, "This one at last is bone of my bones and flesh of my flesh. This one shall be called Woman (isha) for from man (ish) she was taken"." Genesis 2:3). So if we can make this connection then Abish (אֲבִאּישׁ) takes on the meaning "father of man". This may seem a little strange for a female name but this would not be the first time a cross gender name has been given to someone.
This additional speculation may or may not be useful but it does show that Abish can be thought of as a Hebrew name with a specific meaning that may be significant given the story of Abish's father and his conversion to the Lord. To settle this question we would need more information, information that Mormon did not include in the Book of Mormon. Still he thought it significant enough to include her name (or alternately other Nephite record keeps thought her name sufficiently meaningful to remember and to write down). So there may have been special meaning associated with the name Abish which may be why it was the only female name that Mormon himself included in the Book of Mormon.
[Added note (2/8/15): There is an interesting article on Abish recently published over at the Mormon Interpreter.]