I suspect the name just arose naturally (for obvious reasons) and that it would be tough to trace back to any single person. After Cartan-Eilenberg proved it in 1956 (Homological Algebra, p.40) the first mention I see in English is by Tate in 1966/67 (p-divisible groups, p.178) followed by Hartshorne in 1968 (Cohomological Dimension of Algebraic Varieties, p.446), neither of which bother with a citation, reference, or quotation marks (1). However, it was used a bit earlier - also without citation or quotation marks - by Begueri-Poitou in 1965 (2) as 'lemme du serpent'; mentioned early on in their abstract. [NB: the first page of the linked pdf incorrectly lists the second author's surname as Poiton.]
I realize this answer is a bit unsatisfying, but the best I can say is that the name took hold at some point between 1956 and 1965; though I can't even say for sure whether the first use was in English or French. Each of the references above uses the term so casually that I would guess by the late 50s/early 60s it was already being referred to as such in Algebra classes -- though this is just a hunch.
I also did searches in Russian (and, for fun, Chinese) but could find nothing appearing any earlier than 1965.
If I were to suggest where to look next, it'd be in Cartan's Seminar Notes (reference: H. Cartan, Séminaire E.N.S., 1950-1951) or perhaps in the recently published book of letters between Cartan and Weil (Correspondance entre Henri Cartan et André Weil 1928-1991) to see if the word 'serpent' ever comes up.
Edit: I used numdam to search for 'serpent'. Cartan has a quotation about a snake nearly biting its own tail (1965, pdf p.16/17), but more interesting is a paper by Grothendieck dating to 1964 mentioning a snake diagram ("le diagramme du serpent", pdf p.195/258) that he attributes to (Bourbaki, Alg. comm, chap. I, $\S$1, no 4, prop. 2). You can see the term snake diagram in the much later English translation, but I'm not sure when the original French version was publishedwritten (I think at least as early as 1961). If someone could dig up that reference, it would probably hold the first published instance (rooted out thus far) that uses the snake terminology.
My guess for the time being: The term snake diagram originated (in French) around 1960 and was first used by one of the Bourbaki members (possibly Cartan, Eilenberg, or Grothendieck). Snake lemma almost certainly has a similar origin.