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-Eileneberg 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-Poiton in 1965 (2) as 'lemme du serpent' (mentioned twice; first in their abstract).
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.
(For what it's worth, I also tried various searches using different Russian terms for 'snake' or 'serpent', but came up empty-handed; then I did the same in Chinese [蛇引理] just for fun.)