[adastra]

Depends on how much time you have and how big we're talking about. If you have years, or lets say a decade, you'd probably try a bunch of things in parallel. Some strategies could be crazier than others, but you wouldn't waste too much time on things that have huge technological risk. Landing a automated mining robot + mass driver seems far-fetched. As does painting it (for something kilometers in diameter, how would that work exactly?)

So you're probably talking about one or more kinetic impactor missions to try and deflect it billiard ball-style. Before those actually launch, the laser-ablation strategy seems like a pretty attractive option to do in parallel. You'd build an array of lasers that ablate the surface of the asteroid, ejecting material that works as reaction mass. Northrop has 100kw solid-state lasers working right now: http://optics.org/news/1/7/13

Seems plausible that you could get something up and running quickly to see if it would work. Before either scaling it up or dropping back on the riskier impactor missions.

[InclinedPlane]

Ablation is the best bet if you need to move something as much as possible as fast as possible. And for that you want a device which outputs a lot of EM radiation at tremendous power levels and ideally is as compact as possible in order to transport it near enough to the object. Nuclear weapons meet all those criteria and so are likely to be the best tools for pushing around big rocks quickly on short notice. You simply explode them at an appropriate stand-off distance and the energy release of mostly soft x-rays causes the surface near the explosion to ablate, generating thrust.

[JoeAltmaier]

I think we need an asteroid in orbit (or several). Then adjust their orbits as needed to absorb the impact.