Harry and Snape are stuck together on Christmas Eve.

Potential ideas/options:

• They're at Grimmauld Place for a meeting, but Harry has been living there with the Weasleys.  Harry has been quarantined all day because he's been sick and there are so many people there or coming and going, they don't want him to spread it around.  Later that evening, the door opens, and Molly ushers Severus in, also sick, and now also in quarantine.
• Harry has been placed in a safe house for his own protection.  Maybe he's already 18 or 19 and out of school, and has been told to stay put or else.  Snape is brought a few days later, and is told to stay put for his own safety.
• Harry and Snape get locked into a room in the castle on accident and can't get out.
• Lily’s old protection spell draws them both into the same space on Christmas Eve.

Snape always wanted to be nothing like his father. Which is why he was determined never to beat a kid (not that Hogwarts would allow him to). But he can't just snap his fingers unlearn the instincts and attitudes aquired from Tobias Snape and the Death Eaters, that he still sort of believe - that tell him that the strong force others to respect them, that beating those who don't respect you until they just. shut. up. means strength, and that resisting the impulse to attack those who walk all over you is weak and pathetic. In his bones, Snape believes people are either weak or strong, and he's been choosing weakness by keeping his fists off of mini-James Potter while Potter mouths off to him, walks all over him, with impunity, for years.

Another side of Snape knows that it's terrible to beat a child. That normal people probably never want to hit kids. That he will never, ever, become like his father. That he must not indulge that side of himself, that indulging his nasty side is something he has left forever in his Death Eater past (except in little ways that don't matter, like scaring Neville). He will let himself feel weak, even if playing by the good guy rules doesn't earn him much respect, let alone fear, but mostly just earns him scorn for still not being nice enough or good enough who knows what else other than the fact that he won't just beat some respect into them. 

Then Snape finds himself responsible for Harry outside of Hogwarts. Harry is actively antagonizing him. Harry, perhaps, keeps putting himself in danger in defiance of Snape's orders. And finally, Snape decides to beat the fool out of him. Once he's finally done what he's been itching to do for years, what happens next? How can this be the turning point in their relationship in such a different way than Snape wanted it to be?

Harry should probably be of age if they end up friends afterward, so that it isn't some messed up abused child psychology.

There is truth to the "voodoo dolls" [I know the term is problematic] cliche in the wizarding world. It is actually real / applicable, and only Snape knew about that (cue Prince Family Secrets, etc.)

It's the reason why Voldemort only came back many years later, with no nose and not so powerful, while Harry's survival is, in large part, because Snape keeps pouring some low-level luck potion and health potions over Harry's doll.

Takes place during Umbridge's rein of Hogwarts. Harry is in trouble but can't directly communicate his problem because Umbridge and/or ministry officials are there. When Snape gets involved Harry keeps addressing him as "Sir?" clueing him in to the fact that something isn't right. This time he takes appropriate action.