What I've noticed in a lot of the fic I read is that a big chunk of it zeroes in on his relationship with Yukio. In the source material, that's pretty central, but fanfic really digs into the aftermath of 'the reveal' and all the messy, unsaid stuff the show sometimes glosses over. Writers love putting them in situations where they're forced to rely on each other, but without the safety net of the classroom or Shiro. I read one a while back where they got stranded somewhere mundane, like a human-world bus station in the middle of the night, and just had to talk. No demons, no immediate crisis, just the awful weight of their shared past. That kind of story lets Rin's emotional intelligence, which he downplays, actually develop.
Another way I see it explored is through his powers. Canon shows him struggling, but fanfic often asks 'what if he lost control worse?' or 'what if his fire wasn't just for fighting?' I've seen AUs where his blue flames are a curse he has to learn to live with in a non-combat setting, and his growth becomes about acceptance rather than mastery. It's less about getting stronger for the next battle and more about finding a way to exist without burning down his own life. That's a slower, more internal kind of growth that I find super compelling. And it usually involves a lot of Shiemi or Suguro or even Mephisto stepping into weird mentor roles the anime never fully commits to.