I find it intriguing that the best ways to deflect an asteroid from a collision with Earth always involve the Sun's light. NY Times reports on a new (to me) idea of sending a swarm of small spaceships with parabolic mirrors. These mirrors would focus sunlight on one spot on the surface of the asteroid - much like a child using a magnifying glass to focus sunlight on an ant. And just like the ant starts smoking, the spot on the asteroid would start vaporizing, and the gases coming off the asteroid would act as a rocket to push it away from its current trajectory. One benefit of this method that the Times points out is the only difference between a small and a large asteroid is how many mirror ships you send.
My favorite oddball technique is still painting half the asteroid reflective, and half absorptive - the dark side would absorb the momentum of photons from the Sun, while the light side would reflect them, and it would act like a single vane of a radiometer and the trajectory would be changed. The drawbacks to this method are that if the asteroid is spinning the trajectory change would be less, or erratic since the direction of thrust would be continually changing - and pretty much everything spins some. And painting it of course is a pain in the butt.
The mirror ships method has another advantage now that I've mentioned direction of thrust changing - the spot on the asteroid that they aim at can be changed. If the asteroid is spinning, the mirror ships can still always point at the side away from the Sun, for example. If you need to make adjustments to its trajectory, like it's moving a bit too fast, aim for the front side, and when it's going a bit to slow, aim for the back side.