An amusing though obscure fragment of Norse Mythology, Aurvandil's Toe is the only instance I can recall of a star being named after a frostbitten, amputated body part.
Aurvandil doesn't have any detailed stories about him that we know of, but he appears to have been a loyal sidekick of Thor, accompanying the thunder god on various quests and adventures.
Anyhow, Thor and company were returning from Jotunheim (the land of the giants) when they came across the poisonous river Elivagar. A fierce winter storm was raging, and Aurvandil was in bad health so Thor put him in a basket and carried him home across the river, saving him from otherwise-certain death. But one of Aurvandil's toes remained uncovered, and it soon froze. Seeing this, Thor broke the useless appendage off his companion and tossed it up into the sky, where it became a bright star, known to the Norsemen as Aurvandilstá, Aurvandil's Toe.
Thor later recounted this tale to Aurvandil's wife Groa while she was weaving a spell to remove a piece of whetstone embedded in his skull from yet another giant-killing expedition. She became so fascinated that she forgot how to finish her spell, so Thor remains with a rock in his head.
The name Aurvandil was common throughout much of northern Europe. Its Anglo-Saxon cognate was Earendel, often associated with the morning star and most likely the basis for J.R.R. Tolkien's legendary Eärendil, whose ship shone just as Aurvandil's toe.
The exact location of the star known by this name has not survived. Given the story's context, it seems reasonable to narrow the possibilities down to bright stars visible during Northern Hemisphere winters. It might be Sirius.