It's mainly meant as a way to give some personality to the protagonist even if they're silent. I think one of the origins of it could be some of the idling animations seen in the Opposing Force expansion to the original Half-Life. I remember in that game there was an alien weapon that you could use that was technically alive and sometimes Shepard would actually pet the weapon.

Speaking of, that's something that they added into the fan remake of Half-Life 1, Black Mesa. In Black Mesa, when Gordon gets the Hivehand (the one that shoots bee-like projectiles) he has an animation where he's clearly hesitant to equip the weapon on his hand, but he ultimately does it, making it look like he's basically fisting the weapon.

When you want to rip and tear, but gas prices are ridiculous…