This is one of the etymologies that delighted me in my salad days. What could seem simpler? Part of a nail that hangs, right? Incredibly though it’s possible that the nail referred to is an iron nail. Over to OED.
hang-nail, n. < hang v. + nail n.; but historically an accommodated form of angnail
agnail, n. (OED redirects angnail here)
Probably cognate with or formed similarly to Old Frisian ongneil , ogneil , ognīl ingrowing nail, hangnail < the Germanic base of ange n. + the Germanic base of nail n. Compare (with the same initial element) Old English angseta carbuncle, abscess, boil. Compare later hang-nail n.
The original semantic motivation for this formation is unclear. Although all the complaints it denotes cause pain in or around the fingernails and toenails (compare nail n. I.), it is possible that the word may originally (in sense 1) have shown nail n. II., the hard, rounded external callus of the corn (and perhaps also its internal root) being taken to resemble an iron nail driven into the foot. Compare e.g. Old English wernægel wart, tumour on the back of cattle (see warnel n. and discussion at that entry), and, with similar extension of meaning, classical Latin clāvus (iron) nail, also ‘wart, tumour, corn’ (see clavus n.). Senses 2 and 3 would then show subsequent extension to other painful conditions affecting the area around the fingernail or toenail, by association with nail n. 1. However, this argument does not appear to be supported by the senses attested for the cognate in Old Frisian.
With sense 2 perhaps compare also French angonailles (1611 in Cotgrave, glossed as ‘botches, (pockie) bumps, or sores’) and post-classical Latin anghiones , anguinalia , carbuncles (attested in undated saints’ lives).
The forms in n- and gn- show metanalysis (see N n.). The regional variant angernail apparently shows remodelling of the first element by association with anger n.
And there we have hangnail. I am in buoyant mood this fine day as Amazon have just delivered my replacement copy of Ovid’s Metamorphoses, translated in glorious ‘fourteeners’ by Arthur Golding. Shakespeare knew this version like the back of his hand, as many of his lines testify. It’s one of the books I’ve loved with a passion ever since first reading it. And I left it on a bus! (I take books out with me as I often read while strolling along.)
OK, sorry for the digression. Normal service will now be restored.