I want to allow a user to move the position of a skinned mesh's bones by dragging part of the mesh around that bone. Any tips as to how to approach this?
Well, you need a drag & drop script that you can attach to the bone-GameObjects that you want to drag around. The bones need a collider (can or should be marked as "isTrigger"). Meshcollider wont work because it's one big Mesh and therefore only one collider. Use a SphereCollider or box collider then you should be fine ;)
If you need help with the drag & drop script just ask (post a comment). ps. tell us what language you use / want use