1768, "to walk (through or over something) without raising the feet," originally Scottish, a word "Of uncertain and possibly mixed origin" [OED], probably from a Scandinavian source related to Old Norse skufa, skyfa "to shove, push aside" (from Proto-Germanic *skubanan, from PIE *skeubh- "to shove;" see shove (v.)).
The meaning "injure the surface of by hard usage or grazing with something rough" is by 1879. Related: Scuffed; scuffing. As a noun, "a slight, glancing blow," by 1824. Compare cuff (v.2).