"silk-spinning organ of a silkworm or spider," coined 1826, diminutive of spinner with -et.
Entries linking to spinneret
spinner n.
early 13c., "spider," agent noun from spin (v.). Meaning "person who spins textile thread" is from late 14c.
-et
word-forming element, originally a diminutive suffix but not now always felt as one, Middle English, from Old French -et (fem. -ete; Modern French -et, -ette), from Vulgar Latin *-ittum/*-itta (source also of Spanish -eto/-eta, Italian -etto/-etta), of unknown origin. The French forms are reduced to -et in English, but later borrowings of French words in -ette tend to keep that ending.