My cocoa powder settles in the bottom of the mug when I make hot cocoa using half-and-half (the dairy product). Is there any technique to prevent this?
Stir! It's normal :-). If you want a really smooth suspension, you won't be making it with real cocoa powder, or real dairy.
However I always make a paste with the cocoa and a bit of cold milk while the rest of it is heating up. That gives the best texture to the final product. If you don't do that, try it.
The most effective way would be to not use cocoa powder. If you use an instant cocoa drink, you will get a completely dissolved drink. The downside is that the taste is very different and you can't regulate the sweetness. If you use a bar of chocolate (the process is like making a very thin ganache), you will get much less settling, although you will likely have other cosmetic problems such as some cocoa fat droplets swimming on top. A third option would be to cook a spanish style cocoa drink with starch, which is thicker, but better dispersed.
If you insist on staying with just cocoa in milk, you will never completely prevent settling. Still, you can reduce it. First, make sure that you start it properly, as George M said, by making a paste first. Second, cook it up until you see boil bubbles and then let it cool down, instead of just mixing the powder in the warm milk. Third, use a powerful stirring device, such as a mixer or a blender, that will disperse the powder better and it will need longer to settle. Fourth, do not make it in the cup from which it will be drunk, but pour it slowly into the cup, such that the thick part stays in the vessel in which it was made. The last one combines very well with the cooking up and the mixer suggestions, since you don't want to put your cup on the stove, and a normal cup is usually too full for inserting a mixer's beater inside without splatter.