At a loose end

The leading theory is nautical: sailing ships were held together by a huge amount of rigging, and any rope end left untied or "loose" was both untidy and a safety hazard. Keeping crews busy re-splicing and whipping loose rope ends was reportedly standard practice for idle hands aboard ship, giving officers a ready-made task whenever there was nothing else to do. It's a plausible, well-fitting story for a language already thick with nautical idioms, though firm documentary proof tying the phrase's coining specifically to this shipboard task, rather than a more general image of a dangling, unfinished rope end, is thin.