The word chayitz derives from chutz חוץ - outside (also "except"), which also gives us the word chitzon חיצון - external.
We also find a root חצה - meaning "to divide". While I can't find a source that discusses it explicitly, I feel that חוץ and חצה are likely closely related - a fence (chayitz) divides (chotze) and leaves something outside (chutz). From the root חצה, we get such words as chetzi חצי - half and chatzot חצות - midnight. From the related root חצץ - also "to divide, to make a partition" comes the word mechitza מחיצה - partition.
As far as the etymology of חצה, Klein writes:
Phoen. and Moabite חצי (= half), Arab. hazwah (=fortune), OSArab חטי (= favor). These words show that base חצה is connected with {chetz} חץ ( = arrow). Accordingly, the original meaning of חצה probably was 'to divide by casting arrows or lots'.
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