Hi Louise
Clemo is a Cornish name. There is still quite a large contingent in the county of Cornwall though so far I have only managed to trace my clan back to Tavistock area and Slapton in Devon (earliest Clemo date in my file is 1807). There are also quite a few variations in the spelling, Clemo, Clemmo, Clemmow, Clemow, Climo, Clymow, Clymow....
I think the jury is still out on the origin of the name, some think Celtic, a few think Spanish, I have also heard it means "son of Clement"..... take your pick!
Good luck with your search