The Mop is encountered early into the game in the second room of Parlay's Suites. At first the PC imagines it is some aggressive monster, with the art reflecting such, but in reality it is merely a long-haired mop. It is personified throughout the game, though the player can never be sure if it can come to life or not. It has a scar in the "chest" of its body.
The Mop is regarded as a flirtatious female who has no hesitation to displaying affection with the PC. She has the ability to sing and will sometimes sing for the PC. She hates The Diplomat.
Berenice's Hair is a parable of a queen who cut her beautiful, long hair for the goddess Aphrodite, so that her King may return safely from war.
The Eremite claims to have visited Diadem, a star in the constellation Coma Berenices. To wake Berenice from her long slumber, he stole her and plummeted to earth, leaving a crater. When she landed, she ripped open a gash on her chest that would never heal. The Eremite describes as her "wild for him" (implying sexually) and with a beautiful singing voice.
It is very likely Berenice is somehow related to The Mop. They both have wounds in the chest, long "hair", and The Mop sings for the PC from time to time. It could be the Berenice from the parable, but how she became a mop -- or if she really is one -- is unknown.