Clasby is thought to be a name of Viking origin that settled in Yorkshire and Lincolnshire (Giving rise to placenames like Cleasby, Claxby and Claxby Pluckacre.) Over time they moved round the coast to Suffolk, Essex and round into Hampshire. Clasby is thought to mean Klaas's (Klaus, Nicholas)village -by suffix denoting a farmstead or hamlet.
I do not believe there is any connection at all between the names Clasby and Gillespie.. Clasby is two syllables, Gillespie is three.. the similarity is tenuous at best