Conina (also known as Conina the Hairdresser) was the daughter of the hero Cohen the Barbarian and a woman he rescued from being sacrificed. She wanted to be a hairdresser but her family traits and legacy got in the way. She may have a love interest in Nijel.
She was recruited by the Archchancellor's Hat to break into the Unseen University and steal it in order to prevent it from falling into the hands of the sourcerer Coin. She met up with the wizard Rincewind and requested his aid in getting the hat to its true wielder, somewhere in Klatch.
She and Nijel can be seen as opposites: she is the daughter of a famous barbarian hero, forced to be a barbarian hero through instinct but wanting instead to be a hairdresser, while he came from an ordinary family, doesn't have a legacy but wants to be a barbarian hero.
Her name is a female variation of Conin, a reference to Conan the Barbarian, a trait shared by her father Cohen.
The story she gives about how her parents met is identical to a scene from The Colour of Magic, where Rincewind, Twoflower and Cohen rescue a young woman from being sacrificed by druids. She seems to lack a sense of humour, taking Rincewind's joke about her relationship to Cohen to be a statement.