Middle English, from Old English *snyflan "run at the nose" (implied in snyflung "running of the nose"), verb from snofl "nasal mucus;" see snout.
The meaning "be in an (affected) tearful state, utter hypocritical expressions of contrition or regret" is from 1680s. Related: Snivelled; sniveller;snivelling. As a noun, "nasal mucus," from mid-14c. Melville coined snivelization (1849). Middle English had contemptuous term snivelard (n.) "one who speaks with a nasal tone, whiner."