A straw boss is usually a supervisor or foreman of menial work, and the label comes from the farm.
The “big boss” was in charge of the entire threshing crew, whose main task was to harvest the wheat from the chaff, which was straw.
The “straw boss” was in charge of the secondary crew, whose job it was to gather and bail the discarded by-product.
The term “Straw boss” these days means a petty supervisor without any real authority.