This is REALLY easy. All you need to know is:
The link to a random post for your blog is your web address /?random
As an example, the random post address for my blog is http://www.mattg99.wordpress.com/?random
How did I find this out? If you’re logged in to WordPress at the moment you’ll see the usual dark grey bar at the top with the drop down options. Under ‘Blog Info’ there’s a Random Post option. Hover over it in Internet Explorer and the destination URL is shown in the bottom left.
How do you make it into a button? Obviously, you can post the link and people can click on it to get a random post, but that’s not really what most people reading this post will be looking for. You know how to give a picture a link, so the best way to make a random post button is to upload a random post picture and give it a link. (On a self-hosted WordPress site you can edit the HTML, CSS and goodness knows what else, giving yourself a really professional looking button which may even play a sound when you hover over it or something!)
If you want one of these buttons on every post you could save yourself a lot of trouble and just put it in the sidebar. To do this, use the image widget.
My random post button (currently) uses the settings in the picture on the left. 100 by 100 was the best size I could find, and I used a caption rather than a widget title.
I hope that helped! If so, feel free to comment, like, Facebook like, tweet, bookmark, share, subscribe, browse, re-blog (but not copy), and/or vote up!