To handle, touch, or manipulate something with one's fingers.
From 'be-' prefix plus 'finger'. This represents the productive use of 'be-' to convert body parts into action verbs. 'Finger' itself comes from Old English 'finger', from Proto-Germanic origins.
Medieval and Early Modern English speakers created wonderful action verbs this way—'befinger' meant to touch or handle constantly, 'betongue' meant to use excessive words, and 'beeye' meant to look at intently—it shows how languages use prefixes to create meanings faster than inventing entirely new words.
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