If you're not familiar with the terminology, a naked domain is the domain name without anything preceding it. So no www., blog. or anything. Just plain example.com is a naked domain. I prefer this setup to the www. prefix when I can because it makes urls shorter and easier to read. There's still debate about it's impact on SEO.
Fix the "part of the process of setting a preferred domain is to verify that you own" warning
First, create both the www.example.com and example.com versions of your site in google webmaster tools. This is exactly the step I had skipped which led to the confusing error.
Make sure that both versions show up as verified sites in google.
Then, navigate into the www.example.com version of the site and change the site's preferred domain to be example.com.
[caption id="attachment_274" align="alignleft" width="300"] See that both versions of site are verified[/caption]
From there, things are straightforward and you shouldn't see any errors.
Just go into each sites site settings. Tap the site name, then Settings->Site Settings in the top right and choose to have the preferred site domain be naked domain.
For more information and tips about the whole process of setting your preferred site, there are many good detailed posts out there. For example, this one. However, I couldn't find a clear answer to the problem I was running into while getting the "part of the process of setting a preferred domain is to verify that you own" error which led to this post.