For more information, see "How do I add or remove login credentials from my account?" in the Help Center.
My solution for this. If you have your own domain name, then you can use the delegate feature to set up yourself as your OpenID provider. This way if the OpenID provider you are using decides to shut down, or they have some big security breach, you can very easily switch providers without switching your on all sites that use OpenID. All you have to do is set up a page, like http://www.example.com/myopenid.html and use that. Personally, I've set up a subdomain, and just set up an index.html file to do the delegation, so my OpenID looks more like http://openid.example.com. The code for doing the delegate (StackExchange in this case) is below.
<html> <head> <link rel="openid2.provider" href="https://openid.stackexchange.com/openid/provider"> <link rel="openid2.local_id" href="https://openid.stackexchange.com/user/#YOURLOCALID#"> </head> <body> </body> <html>
You can get #YOURLOCALID# after registering with StackExchange by pressing the "Use your own URL to Login" button.
Additional to @Kibbee: If you really have your own domain name, it's easy to be your own OpenID provider. You'll get the warm feeling that the internet stays a decentralized network and are immune to shutdowns of other services.
There are plenty of applications out there which do that. I personally use SimpleID and am satisfied with it (not even a database needed, and has support for multiple IDs).
Login information is top-level; changing your login information on one Stack Exchange site will automatically change it for all other Stack Exchange profiles associated with that account.
You must be logged into your account when you begin this process. (This information is also available in the Help Center.)
Open up your user profile page (by clicking on your user name at the top of the screen) and click the "my logins" link above the bio box on the right hand side.
You should see a list of all of your logins.
To add a new login, click "add more logins..." at the bottom of the box. Choose the new login from the list and follow the steps to confirm.
You can remove your obsolete OpenID login by clicking the same "my logins" link, then clicking "remove" next to the login information you'd like to delete.
If you are adding a Stack Exchange OpenID, you will receive a confirmation email. You must click the link in the email to confirm your email address before the Stack Exchange OpenID will show up under "my logins". Please make sure that you are already logged into your account when you click the confirmation link; otherwise, you will create a new account that needs to be merged with your old one.