Canon: Back button Auto-Focus (BBF)

Here is how I have mine set:

Shutter butt. half-press: Metering start
AF-ON button: Metering and AF start
AE Lock button: AE lock
DOF preview button: Depth-of-field preview
Lens AF stop button: AF stop
Multi-function button: FE lock
Set button: No function (disabled)
Main Dial: Aperture setting in M mode
Quick Control Dial: Shutter speed setting in M mode
Multi-controller: AF point direct selection

Or the above with the exception of the Set button. Have the Set button set to ISO. I could never find that tiny ISO button when I was shooting. Now I can adjust shutter, aperture as normal and then press the Set button to change the ISO all while shooting and all with just a thumb and finger.

or

For normal shooting, with One Shot AF, I prefer having AF the normal way, i.e. on half-press of the trigger button. But I also set the AF-ON button to initiate AF with a pre-registered AF point (Home Point, HP). Then I normally have an AF point "high up" in each orientation pre-registered, and thus I can quickly focus with that point instead, just by pressing AF-ON when taking the picture.

When shooting action, using Servo AF, I also use full back-button AF. In such cases, I use AF-ON to run AF with the currently selected point and set the * button to AF with the pre-registered point. If I need exposure lock too, I normally move that to M-Fn. But since I have various action-related setups stored in the custome setup memory locations in the camera, I sometimes use the M-Fn button to cycle between the different custom setups instead.

I also typically program a key to run AF with a broader selection of AF points than the normal one. Thus if I use a single AF point, I may go to single point with expansion when holding the button programmed to alter the AF behavior. Or go from single point with expansion to zone AF, for example. I also increase the alternate AF point selection a notch, and usually increase release priority a notch too. The idea behind this is that when shooting images of runners coming up close to me, they tend to be difficult to focus track once getting pretty close. The AF point will often land on a solid colored part of their clothes, with no real contrast, so having more points on the target at that time helps. That the points are spread out doesn't matter in this case, since the subject fills a lot of the viewfinder anyway.

But before, at a longer distance, a single point is better, since many points may cause focusing on things near the subject, instead of the subject itself.

Here are a few situations where the benefit is pretty obvious:

You are tracking a runner (Servo AF) towards the finish. To keep tracking, you have to have AF active, but since several other photographers also want shots of this runner, you get bounced around a bit. To then keep your finger on half-press, without neither releasing nor pressing too deep, isn't easy. With back button focusing, you keep AF-ON fully depressed and keep your finger of the trigger button until it's time to fire away. That's a lot easier in such a situation.

You are tracking a runner (Servo AF) who runs behind some obstacles. You still want to shoot, since the obstacles are blurred due to being out of the depth of field, but you can't let the camera focus-track at that time. If you do, it will focus on the obstacles instead. By just letting the AF-ON button go, you can keep on shooting without having AF active, and then resume AF once past the obstacles.

You are shooting someone who remains at about the same distance all the time, so you just want to update focus occasionally. But when you shoot, you want the freedom to compose the image in any way you see fit. But momentarily pressing AF-ON now and then, whilst aiming the AF point to the target, you update focus. Then you let go of AF-ON and can fire at will, without updating focus with the AF point now covering the background.

Other people will have other examples, due to what they frequently shoot, but the principle will be similar.

For example, I was out shooting some birds around my house. It would be ludicrous to have One Shot turned on, so I would walk around with AI Servo. For me to get a bird taking off, I would focus on the bird, then just wait and snap the shot as soon as I see it twitch. So my options are to either half-press the shutter on the branch and wait, or prefocus on the branch, let go of the shutter, then when the bird twitches, hope that the AF that kicks in again keeps the lock, or I can just focus on the branch with the back button, and just wait. The shutter only has to meter and fire the shutter, no AF needed, no buttons being held down.

For the typical One Shot AF photos of non-moving targets, taken at your own ease, there's hardly any advantage at all.

Imagine standing near the finishing line for a sports event, being bumped around by 50 other spectators/photographers, holding your camera with something like the EF 70-200 mm f/2.8L IS II USM mounted. Your task is now to start Servo AF tracking as soon as the competitor you want to take photos of shows up at line-of-sight, then keep tracking while taking an occasional photo now and then. At the same time, you may want to skip tracking occasionally, when other people or objects block the line-of-sight, but an image take could still make sense, since the obstruction will be out of focus. Provided you don't keep on tracking then, of course. Once the obstruction has cleared the path, you want to resume tracking again.
When the competitor comes up close, you want to fire off a number of photos, using high speed continuous drive, at the same time as you want to use a different AF point, higher up in the viewfinder. You may also want to change to a larger coverage of AF points during that moment.

By programming AF-ON to focus with the selected point, * to focus with the registered point and M-Fn2 (or depth-of-field preview) to switch to a different AF setup, you can accomplish all this without having to resort to having to hold a certain button at some intermediate position. It's either on or off in this case. Besides, letting go of tracking and still take photos isn't just difficult the normal way, it's impossible.