If anybody decides to make a movie in which an asteroid is gently pushed slightly off course by mild-mannered people who don't scream at each other constantly, I promise to go watch it. I've become rather inured to explody flicks at this point.
Are you saying you don't want any loud kerblangs?
Yes I am a hypocrite, thanks for pointing that out
If someone wanted to make a true to life film, you'd have the scientists detect the asteroid, predict the collision 10 years in advance, and the rest of the time would be watching politicians debate whether or not a fetus is sentient until the asteroid crashes into earth and extinguishes all life.
But of course the easiest way to "gently push an asteroid slightly off course" that we're aware of is a slightly subsurface detonation that displaces a lot of solid matter in one direction in an explody manner.
I would've thought the most practical way, provided there's enough lead time, would be to attach something like an ion thruster and apply a tiny thrust over months if not years.
Even with enough lead time, displacing as much mass as possible at just above escape velocity is the most efficient impulse transfer. For an asteroid of appreciable size, your tiny thrust would have to be not-so-tiny anyway. (Plus, chances are that a large mirror stimulating the Yarkovsky effect might be more efficient at this, and without the issues that a prolonged contact with a complex spacecraft might bring.)
That direction needs to be planned, or you could just make it worse. It was a big worry that when Bruce Willis triggered the nuke, splitting Asteroid Dottie in half, it would just happen to line up with doing nothing, and all would be for naught.
You can certainly plan the detonation point.
My point was what if you split it in half, and you only succeed in making one half arrive at the impact site slightly faster than the other half, instead of changing either direction at all?
All you need to do is to ensure that there's not only the two halves involved. As long as some substantial mass escapes completely, it basically doesn't matter if the rest is split or not - conservation of momentum guarantees a substantial change in trajectory (at least relative to the accuracy requirements for Earth impact).
Of course you're right if the thing isn't just about to hit earth, any change years in advance is almost guaranteed to help.
This offers an interesting variation of the old timey classic asteroids game where instead of splitting apart and pieces flying in directions appointed by Newtonian physics and the conservation of momentum, the pieces of the hit asteroid try to reassemble according to some model of gravitation or attraction.
Considering the escape velocity for such a body, those fragments would have to be very slow relatively to each other.