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.
Note: 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.