@Baba-Bamidele
To that end, I've been looking around online and in the forum archives, but I haven't found a good, concise guide that explains how the registered names of purebred basenjis work. I understand that the breeder/kennel name will be in there somewhere, and sometimes the individual litter will have a theme (or even a name of its own?). What order do these elements go in? And is the call name added in there somewhere?<<
Okay let me try to simplify it.
The purpose of kennel names isn't much different than your last name... it shows who you belong to, it shows what kennel/breeder the dog came from. Some kennel names are registered with AKC and cannot be used without the owner's permission. ( http://www.akc.org/registration/kennel-names/ ) So you need to go here and make sure the name you want is allowed: https://www.apps.akc.org//apps/reg/namecheck/
Some breeders choose and name every puppy in a litter before they place them in homes, or perhaps allow an owner to choose from a list, or come up with their own name (after the kennel name) or a name starting with a letter or a theme (such as types of guns, weather conditions, star clusters etc). Those help identify not only the breeder, but also the litter.
Okay so your dog isn't from a kennel, so unless you knew the breeder and wanted to recognize him/her, then truly there isn't a need to make up a "kennel name". Like it was said before, some people just use their last names. You could use where your dog is from, or put Africa's Cosette (I just checked, you can use that). So with a pet being registered for possible agility or other type competitions, you can use anything you want. That said, I regretted naming my first dog with my name instead of thinking of a kennel name (which I did with my first litter). So I suggest you make up a list of names, have people look at it and see if you might be missing some awfulness .