Secure remember me function?

[eluser]Adam Griffiths[/eluser]
I am in the middle of developing an authentication library. I have had the login feature for some time now but now I am adding in a remember me function.

I use the term function as loosely as possible, it's not a function just a few extra lines of code. Anyway, it sets a cookie with a hash of the users username. I was thinking of checking for this cookie in the "logged_in" function, and then setting session variables. But then I realised it was open to many security holes.

I need to get around the question "What happens if a script kiddie starts stealing cookies?" They could have access to the whole system.

Any ideas on methods to securely keep people logged in for multiple sessions?

Thanks.