Based on this kink meme prompt:

Increasingly paranoid about the loyalty of his lords, Aerys decrees that each of the major houses shall send a child (the eldest that is not the heir) to be a companion to the young Prince Rhaegar. Thus Ned, Catelyn, Stannis, Cersei, Euron Greyjoy, Elia Martell and Denys Arryn all grow up in the Mad King's Court - how does this change things?

I added Mina Tyrell. Mace had two sisters and we don't know which one was elder, but since her youngest is the same age as Margaery and Mace's other sister has no children mention except that she is pregnant, I decided that she was older. I've also taken out Euron as I've edited, since while we don't know when Maron Greyjoy, Balon's second son, was born, it's pretty reasonable to assume he could have been a year younger than Lyanna. (That would make him eight years older than Asha, and eleven years older than Theon. He would have been twenty-one during the Greyjoy Rebellion then.)

Lyanna does not sound four, so any help would be loved.

ETA: I've made her six now, but she doesn't even sound that.

"Why do you have to goooo?" Lyanna whined as she held on tightly to Ned, and though Ned knew he should tell her to stand up and like a big girl, like a good brother should, all he wanted to do was wrap his arms around Lyanna and never let her go. He loved her the best out of all his siblings and he didn't really care anymore if anyone knew it.

"Because King Aerys wants the second heir to all the Lord Paramount's in King's Landing. It's a great honor. No king has ever done something like this before," Ned told her, somehow the words coming out around the lump in his throat. Lyanna narrowed her eyes. She was only six years old, but somehow she knew when something didn't seem right and nothing seemed right about this, their father seemed too worried.

"I'm coming with you," she said suddenly and stubbornly. Ned sighed. He wished nothing more than for Lyanna to come with him, but if this was truly something more, some sort of show of power for Aerys, he wanted her nowhere near King's Landing.

"You can't come, Lya. You need to stay here at Winterfell."

"Why?" she asked, and Ned quickly tried to think of something to say, something that Lyanna will understand. Luckily, he thought of it.

"Because you need to be the Stark at Winterfell."

Lyanna frowned. "What?"

"Yes! I'm going to King's Landing, Brandon's in Barrowtown, and Benjen will fostered somewhere when he's my age. Father will be here of course, but he could be called away on a moment's notice. You need to be here, to show that the North will always have a Stark like they will always have Winterfell."

Lyanna looked at her brother for a minute before she nodded, seeing the logic. "Fine." Then, wondering new things like she always did, she asked, "Who else is going to be there? Do you know?"

Ned had memorized everything he could about them. "There's seven others besides me. Maron Greyjoy, Catelyn Tully, Denys Arryn, Cersei Lannister, Mina Tyrell, Stannis Baratheon, and Elia Martell. Catelyn, Stannis, and Cersei are younger than me by one, two, and three years respectively, and Maron is five. Elia is older than me by six, Mina by nine, and Denys by thirteen. There was issues with Denys Arryn, because he's not from the main Arryn line, but after the heir Elbert there's no more men from that line. They decided to marry him to Lord Arryn's sister's eldest daughter though and with that marriage, he became second in line. I think she's only thirteen now though, so she's not coming with him."

Lyanna's lips quirked into a smile as she heard Ned recite all this information. "Got it."

Ned laughed. He was glad that she was no longer talking about going to King's Landing, though he wished they could figure out some way to make it work.

Lyanna had another question for him. "When will you get to come home?"

Ned bit his lip. "I don't know. The King never said. We're just going there for now."

Lyanna looked at him with big gray eyes. "Please come home soon."

Ned hugged her. "I'll try, Lya. I'll really try."

"The lone wolf never makes it alone," Lyanna said, tears building in her voice. "That's what Mama always said. We have to be a pack, but how are we going to do that if we're apart?"

Ned couldn't help but wonder the same thing. He hated the thought of King's Landing, the south, these people that he's never heard of but they were about to become something like prisoners together because Ned had heard rumors about the king. He had heard rumors that the king was not alright, that he was not sane.

He had heard rumors that the king liked to burn people and while his father reassured him that they weren't true, he saw the gleam of worry in Rickard Stark's eyes.

Ned saw that there was truth to those rumors.