This is totally inspired from a Headcannon I found at Pinterest. Everything in here are from there, I just made it longer and wrote in some lines. That's all for me. Oh, right: Disclaimer: I do not own Harry Potter, nor any characters in this.

Mr and Mrs Potter was sitting in their private parlour a summer night in July, drinking tea and talking about if they should invite someone over dinner soon, if they should get another House elf, their son James and other things, and thinking of going to bed – it was midnight after all. Suddenly they saw a green light from the room contiguous to the parlour and a thump. Someone had come thought the Floo.
"James! James!" a desperate boy cry was heard, followed by sobbing and another "James!"
Mr and Mrs Potter looked at each other before they both ran out into the other room, which was made only to be a way for Flooing guests to arrive, and in there they found a black haired boy with a big bruise on his cheek who was looking around, looking afraid, and sulking crying for their son.

Mrs Potter gasped when she saw the bruise on his cheek. The boy turned around sharply and his eyes widened when he saw them. The boy recoiled quickly.

Mr Potter squatted and smiled gently toward the boy who looked unsure when they didn't immediately ran to him, angry-looking.
"Hey." He said quietly. "Who are you?"
The boy swallowed and looked down on the floor. He used his arm to dry of some tears from his face before answering.
"Si-Sirius B-Black, sir."

A gasp came from both of the Potter parents this time and the boy – Sirius – recoiled even more, until he stood with his back against the wall and then he slid down so he was sitting on the floor, his sobbing becoming louder.
"I-I'm sorry… I di-didn't mean to… to intrude…" he tried to say between violent sobs. "Ja…James said I-I could c-come anytime… if… if it beca-became bad at h-home… I-I-I co-couldn't hold on another day. I'm sorry – I'm sorry – I'm sorry."
He continued repeating that and Mr and Mrs Potter exchanged terrified gazes over what possibly could have gotten the small boy like this. If the bruise was any indication, it was bad.

"Pl-please don't send m-me ho-home." Sirius begged desperately. "I-I can't… please, I need James."
No. Mrs Potter knew there was no way they could send the boy back to the Blacks if it was they who had done this to him, which it most probably was. You didn't abandon a child like that, even if it wasn't your own. Both she and her husband had been quite reluctant when James told them about his new best friend: the Black Heir. They didn't want their son near that horrible family, and especially not the Heir. Nothing good could come from them. But they hadn't had the heart to tell James No (they never had, no matter what it was about) and knew that James wouldn't listen anyway - when that boy got something on his brain, a miracle was necessary to make him change his mind. But this... this was really not what they had expected. Mr and Mrs Potter's hearts almost broke when they looked at the young child. No, never that they could try to stop James from being this boy's friend now.
Mrs Potter walked to the trembling child who shunned away from her hand and smoothed him over the black hair, still well-styled despite everything.
"Calm down, Sirius." She said with encouraging voice, trying to make him stop cry. "We'll not se…"

"Siri?" a sleepy voice came from the stairs. Everyone looked up and saw James standing there in his Quaffle-pajamas, rubbing his eyes and looking newly woken. He blinked a few times, as if he thought he was still dreaming before looking at them again. "What are you doing here?"

"James!" Sirius shouted relieved. He still sounded afraid and upset and sad, but it was like a weight had lifted from his shoulders when he looked at his best friend. James blinked once again before running down the stairs and actually shoving away his mother, since he knew how sensitive Sirius was with body contact with adults – especially adults he didn't know, he was comfortable with for example McGonagall now – and hugging him tightly.

Sirius hugged back life his life depended on it, and seemed to forget that Mr and Mrs Potter even was there when he felt the calming and reassuring body heat and presence of the person he trusted most in the whole world.
"Mo-Mother. She… she's angry because I-I'm a Gry-Gryffindor." He explained between sobs. "And I have taken it… despite the pun-punishments. B-but today is was too ba-bad. And r-right in front of Reggie too!"

He didn't need to say anymore. James already knew what kind of horrible punishments Mrs Black gave her sons. Especially Sirius now, since he was the family disappointment. He just hugged Sirius closer to show that he was there for him.
Mr and Mrs Potter just stood there, unsure what to do. They had realized they only made the boy – the heir of the Blacks! – even more afraid, and that their son probably was the one that was best on handling him, but they didn't want to leave.

Luckily they didn't have to decide anything, because James looked up at them with expecting eyes:
"Mum, dad, you'll let him stay, right? He can't go back there!" James half asked half demanded.

Mrs Potter nodded immediately and James looked relieved.
"Of course he can stay. You're right, he can't go back there."
"R-really?" Sirius asked, sounding like he didn't believe them. Mrs Potter nodded determinedly and looked at her husband who also nodded.
"Yes."

James continued hugging Sirius until the younger boy calmed down and began to blush in embarrassment that anyone but James saw him like this and pulled away. James grinned when he saw that his friend had gotten back a little of his spirit. As soon Mrs Potter saw that Sirius had calmed down enough so she could take away her son from him without getting a panic attack she took the command.
"James, you're going straight back up to bed. It's midnight and young boys are not to be up this late. Sirius, you can sleep on the couch tonight, and we'll see what we can do tomorrow. Okay?"
Sirius nodded.
"Yes ma'am."
Normally he wouldn't accept sleeping on a couch, but today he was too relieved and grateful that he could stay so he didn't neither want nor dared to refuse.

Mr Potter took James upstairs again and Mrs Potter got a cover to Sirius to have over the night. Normally she would call on one of the House elves, but they were all asleep and she didn't want Sirius to have to meet even more people tonight.
She smiled at him when she told him to lie down and then told him that if he needed something, he could just ask. He nodded, even if she could see that he was still nervous.

Then she left, to give him a chance to sleep. She would check at him later.

When she did come back to check at him half an hour later she couldn't help but smile widely and loving – the worried boy from before were absolutely gone. Instead the boy who lied there looked completely calm and happy. Behind him James lied with his arm thrown over Sirius, holding him close. She had never been more proud over her son than in this moment, seeing him giving his best friend the comfort he needed even if it meant sleeping two persons on a couch.

Mr Potter came up behind her and placed his arms over her waist, holding her close and they smiled together, looking at the two boys.
"They look like they could be twins." She whispered and Mr Potter nodded. There they lied, a big clump of limbs and black messy hair they looked like brothers. It was a wonderful sight. Sirius really wasn't like anything they had expected from a Black.

They looked at their son and the boy who would come to be like a son to them for some moments more before going to bed, letting the boys sleep.

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If you wants to know what made Sirius come over to the Potters in the first place, read Disowned. The answer is in Chapter 2.