This expression was born during WWII as part of the US Office of War Information's attempt to warn people from talking about potential useful information for the enemy. ( source )

A cannon unrestrained on a ship is unpredictable and that's exactly what the term loose cannon means, someone or something that is unpredictable. ( source )

This phrase comes from the cotton industry and dates back as far as the 1780s, when the industrial revolution made mass production of textiles possible for the first time. Huge mills sprang up to take advantage of the new technology (and cheap labour), but it was difficult to keep all the machines running properly. Any machine that broke down or produced defective cloth was said to have a screw loose somewhere. ( source ) How it evolved to the meaning of someone who is unstable is unclear. No research I did traced the evolution.