What is the single word for the tiny pieces of hair left in your razor after you shave?
While I've never heard the word used in practice, several online slang dictionaries list snoodge with this meaning:
The substance (consisting of compressed hair shavings, skin, shave cream, blood, and other facial detritus) that collects between the blades of a multi-blade razor, diminishing its effectiveness.
Next razor I buy, I'm getting the kind with the built-in snoodge ejector, so it won't get clogged up and thrown away so quickly.
This sounds to me more like a sniglet than a slang word in actual use. I would use Gnawme's suggestion of stubble instead if you prefer clarity to cleverness.
It's called stubble on your face, and it's still called stubble in your razor.