Mouse is also boxing Jargon for a welt beneath the eye. These pronounced blood-filled bruises resemble a small rodent due to the "tail" running from the ear along the ridge of the cheek bone leading to a large bulbous "body" nearer the nose.
The term is mostly used in boxing circles but it is also found in most martial arts and contact sports as a mouse is most often caused by repeated blows to the face.
A welt beneath the eye is a severe concern for a boxer. If a Mouse becomes too big it might obscure the pugilist's vision causing the fight to be stopped. Even worse than this, the skin might rip or burst on further impact necessitating plastic surgery and a long absence from the ring.
Between rounds these swellings are often iced by the "cut man" who is employed to keep the fighter in a fit state to continue. Traditionally these welts were vigorously rubbed with ice however this often caused more damage than good.
Modern scientific methods have proven that it is far more effective to apply a cold metal rod called an Enswell* without movement to reduce the swelling.
Sir Henry Cooper almost always developed a mouse during his fights, a disadvantage which Muhammad Ali famously exploited in their fights together.
References
www.thecutman.com
*Enswell is a trademark