I like [name_f]Colette[/name_f] more than [name_f]Elizabeth[/name_f]! Both are great names, but I know many Elizabeths and no Colettes, so that is why [name_f]Colette[/name_f] has my vote.
Ette is a cute nickname, but I am someone who prefers nicknames that start with the same letter as the real name (this is because my own nickname that I always go by doesn’t start with the same letter as my given name, and it is annoying in professional settings).
So as for [name_f]Colette[/name_f] nicknames that start with C, I really like [name_u]Coco[/name_u], and I think that [name_m]Cole[/name_m], Cola, and [name_m]Colie[/name_m] are also very usable