The Hebrew word nāḥāš “serpent” appears to come from a common Semitic root *n/l-ḥ-š (compare Aramaic lḥaš and Geez laḥasa) meaning “to hiss” and “to whisper,” and by extension “to intone in whispers = cast a magic spell.” BDB suggest that the root is of onomatopoeic origin. The related languages have l in the place of n for the most part; the root n-ḥ-š “to whisper; mutter” is indeed found in Aramaic, but most frequently appears in the D-stem and the reflexive of the same. It is therefore likely back-formed from one of the various nouns from this root relating to magic and more specifically divination (e.g. mnaḥḥǝšā or mnaḥḥǝšānā, “diviner;” nuḥḥāšā “divination;” nāḥšā “prophet;” naḥḥāšā “soothsayer”).
Whenever I consider this root, and the semantic extension from “whispering” or “hissing” to “soothsaying” and “incantations,” I am invariably reminded of Sūrat Al-Nās (114), since it not only talks about a “whisperer” but also does so with a hissing sibilance, as you can hear for yourself here. Al-Nās is one of the shortest suwar in the whole Qur’an, and frequently used as an incantation to ward off evil, either recited or written on amulets. In this surah, the worshiper seeks refuge from “the evil of the retreating whisperer (al-waswās al-khannās).” The term “whisperer” (al-waswās) is of transparent origin, but the origin of the following term is less evident. It is not a typical adjective at all, but rather a noun in the faʕʕāl pattern, which is generally reserved for occupations and professions, among other activities in which one might characteristically or habitually engage.
Lane’s Arabic-English Lexicon derives this word from the base form khanasa, which he defines in his customary prolix fashion as “He went, or drew, back or backwards; receded; retreated; retired; or retrograded; or he remained behind; held back; hung back; or lagged behind.” Consequently al-khannās (in the faʕʕāl pattern) would be the one who habitually or professionally hangs back, and this is said to be an epithet for the Devil, as he allegedly shrinks, or retires, or hides himself. This word is identified as an epithet for the Devil precisely because it so appears in al-Nās; in fact, it it apparently has no other referent in the entirety of Arabic literature, and so we are left to consider whether this is not a common noun that came to be applied as an epithet, but rather the convenient etymologizing of a proper noun that is otherwise lacking in a clear or convincing etymology.
Now that I think about it, I wonder whether khannās might not be related to the same root as Heb. nāḥāš, making it a synonym of waswās, with the metathesis of the first two radicals, which is not at all uncommon in the Semitic languages, particularly when dealing with liquids and nasals. At first glance, it does not appear terribly likely, as the equivalent root would have to be laḥasa / naḥasa, with a pharyngeal fricative (ħ) rather than an uvular one (x). Both exist; the former means “he licked (something)” and the latter means “he is unlucky” (and is indeed suggested by BDB as a potential cognate for the Hebrew root), but neither make much sense in this context. On the other hand, if it entered Arabic from, say, Syriac, in which the reflexes of Proto-Semitic *ħ and *x have merged, that might explain the use of the uvular fricative, as this is the usual articulation of this sound among the East Syrians, and indeed we do have a potential candidate: Syriac naḥḥāš(ā), whose stock in trade is whispering, after all. It makes sense, on some level, that an Arabic epithet for the Devil would relate to the Hebrew word for “serpent,” especially given the associations of both words with whispering.
Ultimately, however, this hypothesis risks drifting into Luxenbergian waters, due to the two unconditioned sound changes that it requires. The first is the metathesis of the first two radicals (naḥḥāš > khannāš), and the second is the articulation of the sibilant, which we would expect to be postalveolar (ʃ) rather than alveolar (s). Neither on its own would be particularly fatal, but combined they require too much special pleading to be plausible.
Update: As Nizar Habash points out, though, one of the words for snakes and other such varmints in Arabic is infuriatingly similar: حنش ḥanaš-.