I actually really like the name Clem. My only problem with it is that it might sound too "made up" or "trendy" as my daughter grows up. I really like the name Clementine, too. But that seems too long and everyone else will just think of the fruit. My plan is to name my daughter a different name, but call her "Clementine" or "Clem" as a nickname. I don't want her to be stuck with a name that sounds unprofessional during job interviews or things like that.
Red Skelton had a famous alter ego on his radio and TV show called Clem Kadiddlehopper. Clem is usually thought of as a hillbilly type of name. As in "Clem comes to the big city" a line I've run across a few times.