The header of my blog has links to popular articles and other top stories on my site. Readers find this useful to quickly find articles. Google sees the same content on the top of every blog post and sees it as duplicate content. How can I keep this information at the top of each post, yet have it so that Google just disregards it?
Not sure exactly what you are looking for. But you might want to do a search for "No Follow Tag" on google. That prevents google from following a link. But what you might be looking for is a way to prevent google from reading some of your content. If that is the case, then I have no idea.
There is no problem with your situation. Nearly every website has a block of duplicate content, it's called a navigation menu. Google knows how to deal with this kind of duplicate content. You will not get penalized for it. If you are worried, maybe you should move that menu further down the page. If you use css you could put that menu in a div further down in the code but force it to be displayed near the top.
On the other hand you can redesign your blog so that CSS would take care of placements of the blocks. This way in a pure HTML form your header would appear at the bottom, but with CSS included the header would be on the top. Sounds like a pretty simple issue to me.
In hindsight, probably no but taking them out would break the custom blog theme, that's why I was hoping to hide the links from google...the header has about 20 links to other articles on my blog. I can't make the links no follow, no index because the articles are all original content that I wrote and want in the google index, but the 20 links on each page is causing the blog pages to go into supplemental listings.
If it is just menu links to your most popular articles it is not a problem. Most bloggers have those in their sidebars. Shouldn't really affect your SEO negatively.