A dog that comes up tail wagging with an old toy he or she just found (hopefully not a teddy bear), because he or she obviously wants to play.
Maybe when there is no danger near he finds something like a piece of rope, comes up crouches down in a playful stance. butt high, tail wagging, paws spread in readiness to dart of in any direction and shakes it at you because he likes to tug with you,
They can't talk to but they can come and lie down at your feet, if you get a chance to stop, sit and rest, they sit and lean against your leg, head up, ears shifting in different directions as they listen and watch, speaking volumes without a voice actor. Look up at you with just their eyes, without tipping their head.
Little shows of affection and caring, other characters would not try to convey by word or deed.
They can also show such affection or not to other companions and beings met along the way. Maybe sniff at a ghoul or other mutant who has a dog and "says you must smell my dog", let them kneel down and stroke him and say something a true dog lover would say about dogs.
Probably too late to bring this about in this game, but still a good time to get people to start talking and thinking about such things for DLC, mods, or the next game.