The name Abishai in the Bible
Abishai is the son of Zeruiah, the sister of king David. Abishai's brothers are Joab and Asahel. Before Abishai becomes important in David's army (2 Samuel 10:10), he joins him on his little stunt to the sleeping king Saul (1 Samuel 26:6). Much later he kills a Goliath-like giant named Ishbi-benob and saves king David's life (2 Samuel 21:16-17).
In 2 Samuel 10:10 and all of 1 Chronicles, his name is written as אבשי (Abshai). In 1 Samuel and the rest of 2 Samuel, he's called אבישי (Abishai).
Etymology of the name Abishai
It's not directly clear how the name Abishai is constructed, but there are a few possibilities, and it's also quite clear that the first part of the name Abishai consists of the common Hebrew word אב (ab), meaning father:
But the second part of the name Abishai is disputed. One possibility is that it's meant to be the same as name ישי, which is the name of Abishai's grandfather Jesse. In English these names don't really sound alike, but apparently they did in Bible times. The Septuagint lists these names as Ιεσσαι (Iessai) for Jesse, and Αβεσσαι (Abessai) for Abishai.
To make matters worse: the meaning of the name Jesse is also disputed, but may mean either My Husband or Yah Exists. The name Abishai would thus mean both My Father Is My Husband or My Father Yah Exists.
Another possibility comes when we recognize that in names the word אב is often followed by a yod in connection with names of places to express the lord of a country, city, village. The second part of the name Abishai would then be the masculine noun שי (shay), meaning a gift, or a gift offered as homage (Isaiah 18:7, Psalm 68:30 and Psalm 76:12). This noun is a bit of an orphan; we have no idea from which root or what verb it derives.
The name Abishai may thus mean My Father Is Jesse, as proposed by BDB Theological Dictionary; Father Of A Gift, as forwarded by NOBSE Study Bible Name List; or Father Of Gifts, as per Jones' Dictionary of Old Testament Proper Names.