Secret world leader (shhh) said:
Well, I hate to do this to a random poster, but since it's for a story, make sure it says Rogue when you write it.
That said, without knowing the exact nature of the world you're in to allow for technological development, here's what I say for somewhere around our level of understanding...
The head of the agency your characters are working for explains that they have three nil-chance possible solutions on the board, and the sooner you leave, the better chance you have of implementing any or all of them as needed.
The first is that of a device used to divert, create, or channel vectors of dark matter fields. Dark matter, being a dense material found to have gravitational properties to an extent, actually holds some sway over the stability of star systems. On the notion that the hole's path is directly in relation to what it's sucking on and what has the most or least resistance gravitationally, changing the 'contours' of space might roll this 'boulder' away from Earth enough for it to escape unharmed. Trying to move it 'down' or 'up away from the rest of the Solar System would also be wise. This machine has been programmed with a new DNA computer, so it should be able to calculate and handle the load.
The second is a more dimensional move. Beyond even the dark matter, there is an energetic force which is possibly linked with the dark matter - though scientists are not fully certain of this - which is commonly labeled as dark energy. This is an energy whose approach to normal space is a bit skewed, as an emission which seems to be also important to the stability of space. The device mentioned before has also the proper aparatus to attempt to use these energies to alter the path of the black hole with effects of space itself. More likely, a direct shot wouold fail miserably, as the hole would swallow it up. So, like with the other idea, changing the path or buffeting it indirectly could cause a change.
The last notion is the most dangerous, so much so that it is likely that even with the micro possibility of escape given with this mission, this is almost certainly a death sentence for the crew. Contained within the ship is a device fashioned by CERN containing quark matter, dark matter, anti-matter, and thermonuclear detonator to power the systems that keep it all contained and steady. Using the DNA computer, it has been postulated that a detonation of these unique materials will have some chance of creating the right spark to send the black hole in another direction...IF it is detonated at the spiral of its intake. No direct action of any sort could possibly harm a black hole, but the right forces at the right time have some sliver of hope.
Good luck and godspeed.