The name Cush in the Bible
The name Cush is assigned to two men and two regions. The first region named Cush appears to be situated somewhere in the Babylonian region. Genesis 2:13 has the river Gihon flow around the whole land of Cush, and the Gihon is part of the Euphrates and Tigris (or rather: Haddakel) river system.
The better known region called Cush is usually associated or equated with Ethiopia, but more accurate is Nubia, the region south of Egypt (says the Oxford Companion to the Bible). According to BDB Theological Dictionary, Cush represents the southernmost peoples known to the Hebrews.
In 2 Kings 19:9 and regarding the Assyrian invasion foretold by Rabshakeh, the spokesman of king Sennacherib, king Hezekiah is told to count on the allegiance of Tirhakah the king of Cush. The prophet Isaiah lets Hezekiah know that not because of his alliance with Tirhakah but because of his faith in YHWH will Jerusalem be spared. Quickly after that, an angel of the Lord kills 185,000 Assyrians and king Sennacherib departs and moves to Nineveh. There he is assassinated by Adrammelech and Sharezer.
The ethnonym כושי occurs usually in plural (כושים), to indicate the people of Cush. The singular masculine form is ascribed to a messenger of king David's army, whom Joab sent to the king to tell of their victory over Absalom (2 Samuel 18:21). Another famous Biblical Cushite is the court official of Candace, queen of Cush, to whom Philip witnessed, although author Luke uses the word Αιθιοψ (Aithiops), from which comes our word Ethiopia (Acts 8:27). The Hebrew word for Cushite occurs also as the name Cushi. The feminine form כושית is applied only to the second wife of Moses (Numbers 12:1).
The famous country Cush (and perhaps both of them) was named after the man Cush, the first son of Ham, son of Noah (Genesis 10:6). When Cush insults his father, Noah curses Cush' son Canaan and declares the latter a perpetual servant of his two uncles Shem and Japheth.
Another Cush mentioned in the Bible is a Benjaminite. We don't know anything about this Cush, but he probably wasn't a very nice guy as king David dedicates one of his bitter psalms to him; Psalm 7, "A Shiggaion of David, which he sang to the Lord concerning Cush, a Benjaminite."
Etymology and meaning of the name Cush
The origin of the name Cush is irretrievably obscure, and none of the translators have more to say about it than that it is related to Ethiopia, and having a dark countenance. The prophet Jeremiah rhetorically asks, "Can the Cushite change his skin?" (Jeremiah 13:23), which may or may not suggest that the Cushites were known for being black. Still, this says very little about the meaning of the name Cush.
Klein's Etymological dictionary of the Hebrew Language lists a word written similar to Ethiopia, meaning spindle (with poetic function of 'horn'?), but he gives no applications to try the word. The Septuagint translates this name with a compilation of derivatives of the Greek verb 'to scorch,' and noun 'countenance'.
However, the Hebrew word for black is שחר (sahar). The heth and rosh in this word are so dominant that the name Cush can hardly have come from it. Allowing this would link Cush to pretty much any other word that contains a shin. Like the word ישש (yshsh; weak, impotent, aged) for instance, which makes a far more plausible candidate as a repeated letter often falls away and the yod alternates with the waw. In concert with the common Hebrew particle כ (ke; as if, like), the name would mean As If He Were Weak.
And then there is the root ישה (yshh; meaning uncertain), which yields the noun תושה (tushiya), meaning wisdom, sound knowledge, which would yield the meaning of Cush as As If He Were Getting Smarter.
Still, and for no apparent reason other than a rusty tradition that cost a lot of people their lives and dignity, Jones' Dictionary of Old Testament Proper Names reads A Black Countenance, Full Of Darkness, but also submits the calmly clarifying afterthought, "the etymology is most uncertain." NOBSE Study Bible Name List simply reads Black.
A probably related name is Cushan.