"Booze-catcher" is a colloquial term for a beard, particularly apt when applied to Tigh, who has spent an inordinate amount of time at the bottom of various bottles. Another common variant is "soup-catcher".
Throughout history, men leading insurrections have often worn beards. Though sometimes the wearing of a beard is religiously prescribed by particular factions, it is also a way of distinguishing oneself from the (often clean-shaven) soldiers of the opposing military. They're also sexy, and look good in monochrome reproductions of leaders' portraits.