When we say hallel on days like Rosh chodesh, we don't say parts of two prakim of tehillim. This is often called "chatzi" hallel, or half hallel. In fact, according to http://www.5tjt.com/local-news/13099-our-better-halves the hallel we are left with is closer to 85% of the full hallel. Some siddurim refer to saying the "whole hallel" as "gomrim" we finish the hallel (as in http://www.daat.ac.il/daat/sidurim/mizrah/hol/rosh.htm) , and this makes more sense. The gemara in taanis 28b uses the phrase "gomer bahem et hahalel" and later refers to the shorter version as subject to "medalgin", skipping, but not half (though there might be other places in the gemara which use the word "half")
I understand that "chatzi" does not always mean an exact 50% but I'm wondering when the texts (either liturguical or halachic) began referring to the incomplete hallel as chatzi. Is there a talmudic phrase which would source this (and thus give it semantic value within the context of the gemara's language and use of the word "chatzi") or is it simply a linguistic shortcut introduced later?
The earliest I can find for calling the abreviated Halel "Half Halel" is the Levush (c. 1600):
לבוש אורח חיים סימן תפח
וכשאומרים חצי הלל כגון בראש חודש וחול המועד
Even after him, it is not the standard way of referring to it. Everyone else calls it "Halel BeDilug" which means "Halel with skipping."
Note also that the phrase is used earlier at least to the time of the Terumat Hadeshen, c 1450 (Leket Yosher 1:91) to refer to the two 'halves' of halel said at the seder: one during maggid and one during halel. Possibly the phrase was later borrowed to describe Halel BeDilug.
I can't provide the provenance of the term חצי הלל as sought, but there are examples of חצי's meaning "divided" rather than "halved". E.g., "חצי שיעור אסור מן התורה", which רש״י (to יומא ע״ג ע״ב) explains as meaning "פחות מכשיעור".
The question assumes that chazi in Mishnaic Hebrew means half (which it does in Modern Hebrew). As one of the previous answers mentions, chazi shiur is another place where it seems that chazi means "part of a whole" (or "a fraction") rather than 50%.
I would propose that it is known as Chatzi Hallel even though it is 85% since the 2 parts we do not say are both approximately 1/2 a chapter of Tehilim (Lo Lanu is Tehilim 115 verses 1-11, and Ohavti is Tehilim 116 verse 1-11) which we say the other 1/2 (Hashem Zecharanu is Tehilim 115 verses 12-18, and Ma Ashiv is Tehilim 116 verses 12-19).
If I had to guess, it is probably because the other way "incomplete" things are generally referred to is "חסר" (literally, "missing"), which sort of sounds weak or even deficient or, I dunno, inferior. "Half-Kaddish" is also much more than half of the Kaddish, and I would venture to guess (unless someone can provide a source indicating otherwise) that it is for a similar reason.