This is a branch of another question: What is the best way to implement "remember me" for a website?
The top answer is to implement this: http://jaspan.com/improved_persistent_login_cookie_best_practice
A summary:
Use a random number as a Series Token, and another as a Login Token. Place those in the Stay Logged In cookie, along with the username. Assign a second, normal Session cookie. Each time a user arrives without a Session cookie, consume the Stay Logged In cookie. Issue a new one, this time with a new random Login Token, keeping the Series Token the same.
Why include the username? How is that helping? The Series Token should be enough to identify the user and series. The Series Token was added in this approach to prevent a DoS attack where an attacker just guesses all usernames and hits the site all at once, logging everyone out. But why does it make sense to leave the username in at all?