My friend has poked and prodded and coerced me into posting this šš¦
Let's talk about Jonahbas and where I see it stemming from.
In the one episode we get that is Barnabas' short letter detailing how he ended up in The Lonely(Ep 92 Nothing But Remains), there's a lot of clues to how close the two really are, some from the letter itself but mostly from how Elias (Jonah) talks about it after.
For Barnabas' part, we know that he's seeking Jonah's help because he knows Jonah deals with all this supernatural stuff with frequency. Jonah told him not to cross Mordechai and, fool that he was, he didn't listen. But he refers to Jonah as His Dear Jonah, calls himself his loyal servant. While he might be appealing or prostrating himself in some way out of desperation, to get Jonah to consider helping him, it could also be the plea banking on Affection. Overall we get little from Barnabas to work with.
HOWEVER, the bulk of the support for this ship comes from Jonah and his reaction/behavior after the fact, which we can trust as being genuine, seeing as it's Elias conveying it. We know he did nothing to try and rescue Barnabas from the Lonely because he was Curious. But as we've seen, being an Avatar has an affect on your humanity and empathy. The Eye wanted to watch Barnabas' fate in the Lonely and Jonah was the one who had to do it.
But, as he says, it wasn't because he Didn't Care about him, or lacked affection, which I feel is an interesting and telling turn of phrase. He was Sad when he went to Retrieve His Bones. Now... why would he even do that? That implies that either he personally went to the Lonely to fetch them Or he got Mordechai to get them for him. Meaning that either he probably Owed Mordechai after or called in a favor he was already owed, as something like this would mean Mordechai going out of his way to accommodate Jonah's wishes. Why bother gathering these remains in the first place if the person in question was just an acquaintance? The observation period was over, and a single letter didn't warrent the guilt it would take to go to these lengths. No, he had Affection for Barnabas, he was sad at the loss of this man. More than just an acquaintance to be sure.
Barnabas stated that he has a brother he's very close to, Jonah knows that. So why not give the bones over to him? He would make sure they were properly put to rest. But instead, Jonah holds onto them, keeps them securely in his office. Close to him. And it wasn't as though they were meant to be some sort of morbid display. They're kept in a box Somewhere in the Office, through all those years and lifetimes. And something that stays clear enough in Jonah's mind after all this time that he can recall his feelings after the fact. And perhaps, given the timing, a note to Jon that sometimes Sacrifices are Asked and Must be Given, even of those close and important to you.
So in summary: Jonah and Barnabas were Something. More than acquaintances, more than friends. But the Eye demanded the knowledge through Jonah's own 'curiosity' and so he observed as Barnabas lived and died in the Lonely. Wanting to somehow keep him around, Jonah kept his bones in his office. A reminder of what he's had to sacrifice to make it where he is.