Named after Horn Head, a peninsula abounding in cliffs situated north of Dunfanaghy in Co. Donegal. The modern Irish name of the peninsula is Corrán Binne, meaning 'hook of the cliff'. The English name, Horn Head, is not a straightforward translation. Horn seems to be an approximate anglicisation of corrán.
There is a good case for rendering Horn Walk as Siúlán Chorrán Binne, which was the form adopted by the Northern Ireland Place-Name Project for the street-names project funded by Foras na Gaeilge in 2005-06.