🔼The name Elath in the Bible
There's only once city with the name Elath (אילת) or Eloth (אילות or אלות) in the Bible, and that's the city built or rather repaired by Uzziah (2 Chronicles 26:2) in the territory of Edom (2 Chronicles 8:17). Apparently the city Elath had existed since at least the wandering years (Deuteronomy 2:8).
The name occurs eight times in the Bible, four times spelled with the ו (waw), and four times without. BDB Theological Dictionary proposes that El-paran (Genesis 14:6) is the same as Elath and Eloth, but there's no way to be certain about that either way.
🔼Etymology of the name Elath
The name Elath or Eloth is possibly an ancient version of the more modern (Biblically spoken) Hebrew word אלה ('alla), meaning terebinth. But it may also be a proper feminine plural of the same word:
For a meaning of the name Elath, Jones' Dictionary of Old Testament Proper Names goes with the plural option and reads Terebinths. BDB Theological Dictionary, complete as always, offers both possibilities: Lofty Tree or Trees collectively. NOBSE Study Bible Name List doesn't translate this name, strangely enough.