To make famous; to give fame or renown to someone or something.
From the prefix 'be-' (Old English, indicating causation or transformation) combined with 'fame' (from Latin 'fama,' meaning reputation or what is said about someone). The 'be-' prefix was productive in forming causative verbs in Middle English.
The prefix 'be-' is a linguistic superpower that turns nouns into verbs—'befriend' makes a friend, 'befame' makes famous, 'besmirch' smears—but most of these constructions have fallen out of fashion, leaving us with only a few survivors like 'bewilder.'
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