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][1]). However, it was used a bit earlier - also without citation or quotation marks - by Begueri-Poitou in 1965 ([2][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.

<hr>

**Edit:** I used [numdam][3] to search for 'serpent'. Cartan has a [quotation][4] about a snake nearly biting its own tail (1965, pdf p.16/17), but more interesting is a [paper][5] 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][6], but I'm not sure when the original French version was written (I [think][7] 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.

  [1]: http://tinyurl.com/8h59j7o
  [2]: http://tinyurl.com/9aoxzl7
  [3]: http://www.numdam.org/numdam-bin/search
  [4]: http://archive.numdam.org/ARCHIVE/SB/SB_1964-1966__9_/SB_1964-1966__9__271_0/SB_1964-1966__9__271_0.pdf
  [5]: http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1964__20_/PMIHES_1964__20__5_0/PMIHES_1964__20__5_0.pdf
  [6]: http://books.google.com/books?id=Bb30CjGW7EAC&lpg=PA21&vq=snake&pg=PA4#v=onepage&q&f=true
  [7]: http://books.google.com/books/about/%25C3%2589l%25C3%25A9ments_de_math%25C3%25A9matique.html?id=qqhbcgAACAAJ