I'm trying to create a package which will include a custom field I added to the User object. I cannot find this field in the list of fields that I can add to my package. Is this a restriction? What is the workaround for this?
You can add user custom fields to managed packages - I've done this before and just verified it in one of my dev orgs.
Are you sure that you are seeing all of the custom fields? By default, when you select 'Custom Fields' from the package components, it will show you a subset that can fit on the page. At the bottom of the page, do you have the following link:
If so, this implies there are more fields than will fit on the page. In my case, I have a custom field of 'PIF' on the user record, so if I click the 'P' link at the top of the page, I see the field listed and can click to add it: