Letting this into the world before it becomes debunked by the SE, it's a long held headcanon I have.
The Field Guides mention that Riverstar was the one who created the apprentice system, but this seems to have partially been ignored in later DOTC books like Mothflight's SE, where it appears that mentors and apprentices have already been somewhat invented.
My theory instead hinges on the idea that Riverstar FORMALISED the apprentice system through creating the modern naming system, which is one of the only things that hasn't been created yet in canon (Which is strange, as names are so important to Clan cats). We see in the Cats of the Park that young cats solely have one name, and then pick a second one when they grow up. I'd like to think Riverstar brought this tradition to the Clans, which eventually became our current naming system.
It could have started where there were issues in the Clans with cats not knowing a stranger's rank as all their names are the same, or from some young apprentices who want to change their names now that they're older and found one that suits (Think of Tinycloud during her Warriors ceremony). Riverstar at first suggests the CotP naming idea, but many cats don't want to have "single word" names as it makes them sound like rogues (We see them want to have "Clan Names" in Path of Stars).
Riverstar thus comes up with the suffix idea, with kits being named "kit", perhaps in honour of Emberkit? This could also be Windstar's idea. The apprentice 'paw suffix is created as well, perhaps as a way to show that the apprentice has taken their first "pawstep" into becoming a full fledged Warrior. When a cat becomes a warrior, they choose their second part of their name.
However, some leaders like Skystar (And maybe Windstar) think it's silly for cats to choose their own names, perhaps after some cats choose weird names, or it's a power issue (Which would be in line with both characters). Other Clans, like ThunderClan, want their leader to pick their name as a sign of deference/respect. So they decide that leaders will pick the names with the blessing of StarClan. Eventually this becomes the norm.
Thoughts on this theory? I find it interesting that there's still no explanation for the naming system. Please let me know your own headcanons!