So I have this theory that when Stiles was a baby, Hales used to babysit him. Because the Hales were clearly some of the very influential families in Beacon Hills, and the sheriff being an officer, they must clearly know each other (at least in my head they are)
Of course, Stiles doesn't remember it because he was like one year old. But Derek remembers, so does Peter. But neither of them ever mentions anything because of the whole drama that is happening in the town.
But finally, things have calmed down. Everyone has graduated, and Hale House has been renovated. Peter is sane and forgiven, and everyone is having a pack night, and somehow the conversation is about how much of troublemakers they were as kids, and suddenly everyone wants to know what kind of a baby Derek was. Of course, Derek refuses and threatens everyone with his eyebrows and then leaves the room, so they turn to Peter. And Peter is like, 'Oh, he was such a big crybaby and mama's boy.' and everyone is like 'NO WAYYYY.' Then Peter proceeds to tell stories of baby Derek, who would cry while watching Bambi and wouldn't let anyone change the channel. Everyone laughs at that, and then Peter looks at Stiles and says, 'That was why he used to call you his Bambi and carry you around all the time, never letting anyone else hold you.' The room fell silent after that, everyone confused and surprised, wondering what the hell Peter was talking about. "Oh, you probably don't remember, but we used to babysit you," and everyone is like 'No, you're just trying to pull our leg' and boos him. But Peter is like, 'No, we really did, Talia was close with your mother, so she dropped you here sometimes.' when stiles still denied it Peter went and came back with an album saying he has proof.
And among the very few photos they managed to save was a photo of eight-year-old Derek sitting on a couch holding baby stiles. Derek had this full-tooth smile, and Stiles was in this adorable brown onesie, nibbling on a wolf plushie's ear as they both looked at the camera.
Everyone is so surprised to see the photo, especially Stiles, who has no recollection of it. But he remembers that plushie, which he still has. Peter went on to say that Stiles liked Derek the most and would only calm down when in Derek's arms. How Derek will fight with Laura whenever Laura says it's her turn to watch over Stiles, and how that plushie was Derek's favorite and he only shared it with Stiles, and how he didn't even let Cora, his own baby sister, touch it.
Then Derek walks into the room, and everyone bombards him with questions and the photo. Derek chases Peter around the house for telling them all this. But Stiles is a bit silent, trying to digest what he heard. When he goes back home, he digs out that plushie. Then he asks his dad about whether they knew Hales, and Dad tells him how close they were and how much they supported Stillinski's when Claudia fell sick, and how devastated he was when they died.
This kind of changes how Stiles used to look at Derek. He brings the plushie to Derek and asks whether he wants it back. But Derek is like no, he gave it to Stiles because he wanted him to have it. And upon Stiles' prodding, share some of the childhood memories.
In my head, this will bring them closer, and it is one of the reasons Derek always had such a high tolerance for Stiles' antics and protected him all the time.