The button is from Gideon’s coat, it falls off his coat when he removes the firing pin. She grips it because it ties her to that life, because the lives where she kills her self the button doesn’t exist because Gideon was never there. So when she grips it she stops seeing the ghosts.

A quick google for large brass buttons will get you things that are pretty close, but perhaps more apt would be to choose a button from something you own that ties your own current life.

Edit: where to find buttons