"Half Lambert" lighting is a technique first developed in the original Half-Life. It is designed to prevent the rear of an object losing its shape and looking too flat. Half Lambert is a completely non-physical technique and gives a purely percieved visual enhancement and is an example of a forgiving lighting model.
To soften the diffuse contribution from local lights, the dot product from the Lambertian model is scaled by ½, add ½ and squared. The result is that this dot product, which normally lies in the range of -1 to +1, is instead in the range of 0 to 1 and has a more pleasing falloff.
In the original Half-Life, where this diffuse Half Lambert term was the only term in most of the model lighting, it kept the characters in particular from being lit exclusively by a constant ambient term on the back side with respect to a given light source.