Most autofocus systems need a fair amount of light to operate, and in general they focus faster with a lens that is faster - i.e. has a wider maximum aperature. Sports photographers will use a 300mm f/2.8 lens even if they are shooting at f/5.6 to f/8 because the faster lens acquires focus faster than a 300 mm f/4 or slower lens. Of course a 300mm f/2.8 lens is also typically built for the task, with special focus motors, image stabilization and other features which contribute, but the raw light gathering power is an important factor.
My question is whether this effect works out for the lenses we typically use underwater?
In general the answer would seem to be yes, but the raw light gathering power might be balanced by other factors. Wide angle lenses may not have the same sort of focusing motors as say a 300 mm f/2.8.
A case in point in the Canon 85mm f/1.2 lens. It isn't wide angle, and isn't really suitable for UW use but it is a fast lens. The focusing motor on that lense is ridiculously slow. Canon came out with a second version of the lens that improves the focusing speed, but it is seems to be slower (at least anecdotally) than the 70-200mm IS lens at 85mm.
An as example consider the Canon 35mm f/1.4 and 24mm f/1.4 II lenses. They are two stops faster than other ways to reach those focal lengths which are at f/2.8 (say, the 16-35mm f/2.8 zoom, or f/2.8 prime lenses). Do those two stops matter for either speed of acquiring focus, or the ability to use autofocus in lower light levels? They certainly ought to, but I have not tried them yet. I wonder if anybody else has?
Focusing speed is one issue, ability to autofocus at all is probably more important for UW use. However it is governed by similar factors.
In principle this should also depend a bit on the camera. The forthcoming Nikon D4 and Canon 1DX are supposed to have greatly improved autofocus.