Thanks for the feedback, everyone!
Kincub sounds nice, as would Coalcub. (Coacub doesn't sound right.) I have another name possibility - Cubburn. I don't really like it though. By the way, it's very easy to change the name, so you don't have to worry about me changing it.
Anyway, I'll post some more details about the fighting styles.
There are six styles (not including no preference). They are: persistent, all-out, careful, tactical, instinctive, and calm. The first four have an effectiveness square - persistent beats all-out, which beats careful, which beats tactical, which beats persistent, returning to the start. The other two are supportive, with calm being more of an asset to defensive Pokémon, and instinctive to offensive.
The styles are relative to the average Pokémon of that species, so a calm Primeape is possible, as is an all-out Shuckle or a tactical Bibarel. Just don't expect them to be any good.
Also, the styles aren't exactly a new mechanic, just a different view on an older one. It might help to know that a trainer wanting a specific style should fight species using that style so their Pokémon can learn it, for example fighting an Alakazam to improve your Pokémon's tactics, or a Gyarados to improve "all-out"-ness.
No prizes for guessing what mechanic it describes :P