What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.
I suspect the name just arose naturally (for obvious reasons) but 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.]
Edit: I used numdam to search for 'serpent'. Cartan has an unrelated quotation about a snake nearly biting its own tail (1965, pdf p.16/17), but actually relevant 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, and (confirmed by the comments below) it is found in the 1961 original, as well. As far as I know, this is the first published instance that uses the snake terminology.
My guess for the time being: The term snake diagram originated (in French) around 1961 and was first used by one of the Bourbaki members (possibly Cartan, Eilenberg, or Grothendieck). Snake lemma almost certainly has a similar origin.
If you Google for "diagramme du serpent" it becomes plausible that it was a diagram in Cartan-Eilenberg first of all, before a lemma. Interesting example of how Bourbaki became the standard grad student syllabus.