I could imagine some engineer notices that some more distance probe from the star is suffering from clock skew, but nobody takes them seriously, putting it down to a fault in the circuitry: "It's old, the crystal timer is well out of spec". Then another distant probe also suffers a similar problem, and more people begin to take it seriously.
By the time they collect enough evidence that this is not by chance, other people have started to notice that stars have been vanishing behind a space in the sky. When asked why they didn't notice this sooner, there was no telescope pointed there for a while due to its position and budget cuts to the astronomy programme have limited search time.
Then the real drama begins, they know that amateurs will also notice soon enough, so they need to control the spread of information. They then need to somehow convince people who will most certainly die, along with everybody they know and love, and work towards the common goal of saving the human race.
Then some awesome science fiction around space travel that can escape black holes. Of course something will happen and the escape will be narrow, with them needing to sling-shot off the black hole or something.
I would call it something simple like "From Darkness" as not to spoil too much the story. For some reason Matthew McConaughey will be a space cowboy pilot, Amy Adams as space-travel Science officer, Matt Damon is chief potato grower and Denzel Washington has for some reason got to make a super hard decision to leave somebody behind. Samuel L Jackson will at some point get pissed off and force the launch. In a side story, Bruce Willis starts an expedition to go mine the black hole, and instantly gets noodled.