If I may, your recollection is slightly off.
The finger snapping is started by the character portrayed by Jim Parsons in response to President Kennedy's moonshot speech.
His comment is
"I suppose we should thank President Kennedy for the continued job security"
The assembled scientists / mathematicians indicate their approval by finger snapping also.
Kevin Costner finishes up by praising their work and applauds them and they respond in kind.
Now this doesn't actually answer your question but finger snapping as a sign of appreciation and as a substitute for applause has roots in many ancient cultures but rose to prominence in the US in the 50's 'beatnik' culture.
The reason behind this (as legend goes) is you can't clap and hold a drink at the same time and that it is a quieter form of appreciation than hand clapping.
Now Hidden Figures takes place in the early 1960/61 IIRC so finger snapping would have been more mainstream by then...even if not common.
Although our scientists are hardly beatniks it's possible that this form of alternative applause is just being used as quieter and less disruptive during announcements.
On the other hand, as pointed out by our OP
Jim Parsons was holding a newspaper, so he couldn't clap, which could have motivated his character to snap with one hand. The others would have picked it up as being a recognizable, alternative form of applause.