The boneCapture attribute refuse to be renamed by the Attribute Rename node.

Edit: In the FBX-file those names are gone, so they should be coming from somewhere else. The skeleton has been renamed and that works.
(Note! This was my wrong assumption while just checking the long file manually, see below)

What I'm trying to do is to keep the capture weights and simply have the new name. The capture is possible to store in the FBX and transfer back. However, I've had to repaint parts previously and would like to be able to change the names or capture regions without having to repaint them. Repainting the eyelid weights is not so fun because they need to be very precise.

Attribute Rename just renames the attribute itself, not the attribute values
It sounds like you want Attribute String Edit, which can be used for index-pair attributes like boneCapture in addition to regular string attributes
There's a note at the top of https://www.sidefx.com/docs/houdini19.5/nodes/sop/attribstringedit [www.sidefx.com] for how to edit the capture region names for boneCapture