A short growth of beard, aimed to affect a rugged masculine or deliberately unkempt appearance.
Stubble has been the word used since the 13th century to denote the short cut stumps of grain-stalks left in the ground after harvesting. Since the 17th century 'stubble-beard' has been used also to refer to the short growth of men's beards.
One of the most used phrases coined in the the 1980s
During most of the 20th century a clean-shaven appearance was considered smart and hygienic in Western society. In the 1980s the beard growth that had previously been derided as 'five o-clock shadow' became the fashionable 'designer stubble'. This was the name given to the rugged macho look that men obtained by going a few days without shaving. The image that was conveyed was that the wearer was either too busy or too rebellious and carefree to bother with shaving. It also harked back to the cool and macho appearance affected by actors like Clint Eastwood in his 1960s spaghetti western roles.
The best known of all 'designer stubble' wearers was George Michael. The look wasn't easy to maintain as it required a few days' beard growth and shaving spoiled the effect. The electric beard trimmer came to the rescue and once it became easy to maintain the look for long periods it became much more popular. The earliest reference to the phrase that I have found is in Familiar Friends - Northwest Plants, Rhoda Whittlessey, July 1985:
"De Ijsbreker Musiekcentrum is a vibrant, cutting-edge venue favored by intellectuals and the designer-stubble set."
That reference is to a performance venue in Amsterdam by an American author, so we can reasonably assume that the term was current in the US by 1985. An unambiguous definition of it comes soon afterward's Campaign Magazine, August 1986:
"The beard trimmer, which allows the user to wear a fashionable 'designer stubble' look without having to shave and grow it every three days, is already popular in the US."