🔼The name Ithrite in the Bible
There are two men in the Bible with the epithet Ithrite and we don't know what to do with that. An Ithrite might be someone descending from any of the many Jethers (יתר) but more likely is that they come from a town named יתר, and that would be Jattir (Joshua 15:48). The men called Ithrite are: Ira the Ithrite and Gareb the Ithrite, both among David's mighty-men (2 Samuel 23:38, 1 Chronicles 11:40). On the other hand, these two Ithrites may very well have come from Kiriath-jearim, which housed a whole family of them (1 Chronicles 2:53).
🔼Etymology of the name Ithrite
The name Ithrite may be a regular adjective that says something about the men that carry it as a pseudo-name (without referring to their place of origin), but any Hebrew audience would probably have figured that Ithrite is an ethnonym (that's an adjective created by adding a י (yod) to the end of a name) derived of the verb יתר (yatar) meaning to remain over, be a rest:
For a meaning of the name Ithrite, NOBSE Study Bible Name List reads Preeminence. Alfred Jones (Dictionary of Old Testament Proper Names) does not interpret this name but points towards his article on the name Jether (which unfortunately fails to refer to the Ithrites). But for the name Jether Jones reads Excellence. BDB Theological Dictionary doesn't interpret our name but does refer to Jattir. The name Jattir, however, BDB also doesn't interpret, and declares the root dubious. This may be technically true but not a single member of a Hebrew audience would care much for technicalities, and without second thought tie our name to the verb יתר (yatar) meaning to remain over.