To be fair the article summary above makes it sound like a computer glitch, so if you just went with the slashdot summary and quote I can understand the confusion. From the article it indicated that there was a sensor malfunction (not necessarily a computer malfunction) which means it couldn't autonomously point itself at the sun anymore. Leading to the solar panels not getting enough direct sunlight and thus draining the battery. The computer startup just means it will then consume power more quickly than the solar panels can produce it at the wrong angle to the sun and the battery will drain in a couple minutes again and the computer will automatically shutdown.
Really all we are talking about is whether the computer can boot up quickly enough and whether they can send something like:
1010 Fire thruster X for 23 milliseconds
1020 wait 500 milliseconds
1030 Fire thruster y for 22 milliseconds
(My BASIC is a little rusty though ;)
Assuming they know a precise orientation of the craft when they send the commands they should be able to at least point the craft more towards the sun. Maybe not 100% optimally, but enough to get net power to the computer and maybe begin to charge up the battery. Of course without more information from the computer they probably don't know much about the state of the systems. It could just not work if there are more malfunctions, so there is substantial unknown risk mitigated by the fact that they have already lost the use of the probe so they have everything to gain from the success of a best attempt.
This is all about observation, timing, communication, making some educated guesses, keeping the execution simple to keep it within the estimated window of opportunity and a lot of triple checked math to come up with the correct numbers to send based on all the available information.