It depends on the kid honestly. I have heard most parents say that the pen hurts more since you can't control the injection speed. But that was a plus for one of my daughters and a minus for the other.
My older daughter used the pen initially, with the old formulation of Humira than BURNED. She hated the shots, but refused to use the syringe because she didn't want to see the needle. She also starting doing her own shots soon after starting Humira and the pen was easier.
When she was older, at 19-20, I read on here that the syringe hurt less and got her to try it. It was still the burning formula of Humira but she loved the difference! Said going slower made it hurt less.
My younger daughter also started with the old formulation of burning Humira. Started with the pen and did her own shots at 12 (almost 13). She tried the syringe later and hated it. Said if you inject slower, it still burns and it prolongs the burning. I was shocked at her response since I'd always heard the syringe was less painful.
But now I have seen a number of parents on juvenile arthritis facebook groups say that their kids LOVE the new citrate free shot. That it's incredible compared to the old burning kind. Plus, I have also seen parents there say that their kids actually prefer the pen - usually for the same reason, it's over quicker and it gives the child control by doing the shot themselves.
I've seen kids as young as 7 giving themselves their own Humira shots (with supervision). The pen is much easier, but my older daughter says the syringe is pretty easy too once she got used to it.
So based on my two girls, I'd say it is trial and error. Ask your daughter what she'd prefer. If seeing the needle would bother her.
Also, my girls had no issue with the click when you press the pen button - it's not that loud. They never even mentioned it. I've also never seen any JIA parent mention it so far, but it does make a clicking noise and I could imagine it might bother a kiddo with a sensory processing disorder, for example.
But it's not something we ever had any issues with.
**Edited to add - just read your second message and it sounds like she's on the pediatric dose which is only available in the syringe. How much does she weigh?