From Old English 'nahwaether' (not which of two), from PIE *ne (not) + *kwoteros (which of two).
Not either; not the one nor the other of two things or people.
From Old English 'nāhwæþer' or 'nawþer' (neither, not either of the two), a contraction of 'nā' (not, no — from PIE *ne, the fundamental negative particle, one of the oldest morphemes in the language) + 'hwæþer' (which of two, whether). Old English 'hwæþer' comes from Proto-Germanic *hwatharaz (which of two alternatives), from PIE *kʷoteros (which of two), formed from the interrogative root *kʷo- (who, which, what) + the comparative suffix *-teros (denoting a binary choice). The same formation produced Latin 'uter' (which of two — 'either'), Greek 'póteros' (πότερος, which of the two), and
The word 'neither' is a contraction of Old English 'nā-hwæþer' (not-which-of-two). Its positive counterpart 'either' comes from 'ǣghwæþer' (each-of-two). And the question form 'whether' comes from 'hwæþer' (which of two) — all three words contain the same ancient 'which-of-two' root, but with different prefixes: negative, distributive, and interrogative.