Old English (West Saxon) næddre (Mercian nedre, Northumbrian nedra), "a snake; the Serpent in the Garden of Eden," from Proto-Germanic *naethro "a snake" (source also of Old Norse naðra, Middle Dutch nadre, Old High German natra, German Natter, Gothic nadrs), from PIE root *nētr- "snake" (source also of Latin natrix "water snake" (the sense is probably by folk-association with nare "to swim"); Old Irish nathir, Welsh neidr "snake, serpent").
Since Middle English the word has been restricted to use as the common name of the viper, the only poisonous British reptile (though not generally fatal to humans), then by extension it was applied to venomous or similar snakes elsewhere (puff-adder, etc.).
The modern form represents a faulty separation 14c.-16c. of a nadder into an adder, for which see also apron, auger, nickname, orange, humble pie, aitchbone, umpire. Nedder is still a northern English dialect form. Folklore connection with deafness is via Psalms lviii.1-5. The adder is said to stop up its ears to avoid hearing the snake charmer called in to drive it away.