斤 is a drawing of an axe — you can still see the blade and handle. Axes were traded by weight, and the character slid from the tool to the unit: one 斤, the weight a market scale was built around.
Chinese markets still price everything per 斤 — exactly 500 grams since metrication, so two 斤 make a kilo (公斤). Buying fruit means asking 多少钱一斤? 'how much per jin?' The axe lives on in characters like 近 and 新, where it lends its sound or its chop.
An AXE that became a market WEIGHT — half a kilo per chop.
半斤八两 'half a jin, eight liang' means 'six of one, half a dozen of the other' — because the old jin had sixteen liang, so half a jin literally was eight.