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What is the cognitive meaning of the root כפר? It's applications (that I know of, feel free to add) are:
Is there a common denominator? (I left out usages that I see as synonymous, such as bribe [similar to replacement]).
As @DanF pointed out in the comments, the root meaning appears to be 'to cover', in a literal sense ('to smear') or a metaphorical sense ('to replace', 'to atone'). Gesenius and BDB say as much.
Thus you also have kaporet (Ex. 25:17) which covers the ark, or kopher, the henna plant (Song of Songs 2:14) which is made into a paste and smeared over hands and feet, or kephor, the frost which covers the ground (Job 38:47). Kfar as village probably originates with a cluster of protected or covered dwellings; kfir, they suggest, is a shaggy young lion (i.e. covered with hair or mane).
Cognates of this root with similar meanings are well-attested in Ugaritic, Assyrian, Akkadian, and other languages of the ancient Near East (see, e.g., Lyonnet).
