From Old French (13th century), from Greek 'blasphēmeîn (βλασφημεῖν)' ("to speak evil, slander").
To assign responsibility for a fault or wrong; to hold accountable; to find fault with.
From Old French 'blasmer' (to rebuke, reprimand, condemn), from Vulgar Latin '*blastemāre,' alteration of Late Latin 'blasphēmāre' (to blaspheme, to revile), from Greek 'blasphēmeîn' (βλασφημεῖν, to speak evil of, to slander). The Greek word is from 'bláptein' (to injure, damage) and 'phḗmē' (speech, utterance). So 'blame' is literally 'injurious speech' — the same word that gives us 'blasphemy.' The softening from 'blaspheme God' to 'blame someone for a mistake' is a dramatic deflation, like using a word for sacrilege to describe forgetting to buy milk. Key
'Blame' and 'blasphemy' are the same word at different stages of wear — both from Greek 'blasphēmeîn.' English borrowed it twice: first through French as 'blame' (everyday fault-finding, heavily eroded), then directly from Latin as 'blasphemy' (religious offense, kept intact). Same Greek word, two completely different register levels.