A couple of observations that the media often misses:
If you start when the object is far enough away, the force needed can be quite small. This is very useful for that big rock that just had a near-miss and is likely to hit two or three orbits down the road. Not so good for the one heading straight at us that we didn't see coming.
With most methods for applying force, there is a chance of breaking up what might be a pile of rocks held together with microgravity. One big rock into ten smaller rocks is bad. One big rock into millions of smaller rocks small enough to burn up on re-entry is good. The one method that won't cause a breakup is a gravitational tug; a mass that is placed next to the object and attracts it with gravity, using it's own propulsion to keep it at a constant distance. Alas, the force is very small so you have to start very early.
Given the above, an ideal system would have:
[1] A really good capability for seeing these thing coming when they are still very far away. [2] A gravitational tug with a lot of delta V so it can meet up with and match velocities with an object far out, and a reliable long-term small-thrust system to use once it gets there. [3] A close-in method that can blow big rocks and big slush balls into a very large number of very small objects. [4] Some method for dealing with smaller threats - rocks just barely big enough to destroy a city. Large lasers, (maybe space-based) perhaps.