The Collaborative International Dictionary of English v.0.48:

Hangnail \Hang"nail`\ (-n[=a]l`), n. [A corruption of agnail.]
   A small piece or sliver of skin which hangs loose, near the
   root of a finger nail. --Holloway.
   [1913 Webster]

WordNet (r) 3.0 (2006):

hangnail
    n 1: a loose narrow strip of skin near the base of a fingernail;
         tearing it produces a painful sore that is easily infected
         [syn: hangnail, agnail]