I've created several weapons now in NifSkope, and they all have the same problem. The first person version of the weapon shows up fine, while the third person doesn't show up at all. Does anyone know why that might be?

did you set a model for the weapon? There are 2 slots. The first person/world model which is set up as a static object, then the normal weapon model which is used as the 3rd person.

Yep, I did. Some of my weapons don't show up there. (I think those are because of another issue though. For some odd reason Bound weapons don't show up in my CK as visible, even though they're around ingame. I've been wondering about that, it's weirding me out.)
The problem is exclusively when I use my own meshes. Do I need to add a texture?

I should remark: By "made in NifSkope" I mean the following. I flew by the seat of my pants deleting all the nodes in Dawnbreaker except for the glowy bits. Then I started copy pasting it all over the place, making it into cool shapes. I kept all of the relevant weapon data though, they still swing and collide and everything. They're just invisible unless they're in hand. Clearly there is some difference between a 1stperson mesh and normal mesh that I'm not getting, and I need to change something.