🔼The name Elkoshite in the Bible
The ethnonym Elkoshite occurs only once in the Bible and is a bit of a mystery. It's ascribed to Nahum the Elkoshite, who wrote the book of Nahum, and would technically denote someone from Elkosh.
No place called Elkosh is mentioned in the Bible, but there is a town called Capernaum, and that name means Village of Nahum. It is of course perfectly possible that Elkosh was renamed Capernaum in honor of its famous son, but that too is not told.
🔼Etymology of the name Elkoshite
It's not clear how the name Elkosh(ite) is formed, but the standard approach to a name like this is to assume that the אל part denotes El, the prominent Canaanite deity whose name became applied to the God of Israel, or the common abbreviation of Elohim, the genus God:
The second part of our name looks like it might have something to do with the verb קוש (qush), meaning to lure or lay bait:
For a meaning of the ethnonym Elkoshite, Jones' Dictionary of Old Testament Proper Names proposes God My Bow, which seems plausible in a technical sense but a tad hard to explain practically.
Neither NOBSE Study Bible Name List nor BDB Theological Dictionary offers their suggestions to what this name might mean.