You can override the colour and linetype, no problem. But no matter what you try: Object Styles, Visibility Graphics, Revit Link Visibility overrides, none of this overrides the actual Lineweight of Generic Annotations in the linked file. It is using the value from the Line Weight settings, Annotation Line Weights tab IN THE LINKED file to draw these items in the host file. If Pen 1 is set to 0.3mm in the Link, you will never be able to get a lighter pen weight override in the Host.
To me, this is a bug. If the Revit Link visibility is set to Object Styles - By Host File, annotation line weights should be overriden to match the applied pens.
Unfortunately, this is not the case. Here is a video showing the problem:
Upon further investigation, nested Annotations in Component families that live in a Linked File can't really be overridden by Visibility/Graphics, unless you edit the Revit Link visibility. Even then, you can set line weights and colours but they will still reference the Linked File Annotation Line Weights.
Fairly nasty stuff, particularly if you are working with linked files from consultants / contractors that extensively use nested Generic Annotations inside their families (sidenote: I'm pretty sure this always a bad idea. Model in 3D, use tags for text, if you have to have text in the family use Model Lines or Model Text or some other method so that you don't have nested Generic Annotations. They scale wildly around and become a crutch for not modelling in real world scale).
Ok, so we have a problem. I discovered that there is one master switch that works in the Host file to get around this.
1) Set the Family Category to Halftone, this will override the linked, nested Generic Annotation
2) When printing, tick the "Replace halftone with thin lines" box in Print Setup
I realise that this is not WYSIWYG, and I really don't like that. But it does work. Here is a video:
But wait, there's more. We can actually make it closer to WYSIWYG (is that a thing?), if we adjust the Manage - Additional Settings - Halftone/Underlay to 100. The lineweight will still display incorrectly, but it will print correctly, and there will be no halftone effect.
Some further reading:
Linked File Line Weights - The Revit Clinic