'Jump the gun' derives from track and field races and was preceded in the USA by the phrase 'beat the gun' (or pistol).  The earliest citation for the figurative usage, as we use the phrase today is from The Iowa Homestead, November 1921: "Give the pigs a good start; jump the gun, so to speak, and get them on a grain ration before weaning time." The figurative meaning is that something is started before the preparations for it have been completed. 

The phrase originated as 'son of a military man' (that is, a gun). The most commonly repeated version in this strand is that the British Navy used to allow women to live on naval ships. Any child born on board who had uncertain paternity would be listed in the ship's log as 'son of a gun'. Read more about this phrase and find out if the Navy allowed women on board here . Today the phrase means the boy is naughty.