Sometimes props are commonplace things, but they're useful to set a certain reality for a character. Even if it only plays a small part in a scene, it grounds the character to that scene, and gives the actor something to hold, touch, relax into a scene.
I always compose a props list before I shoot, and the day before I go through it, making sure I've added everything to the set.
It's the little things that make a character seem perfectly at home, and tethered to the events occurring in the story.