This gets a bit more complicated when you're trying to add a Stack Exchange OpenID, particularly if you're already using a different SE OpenID. You can't change the email associated with an existing Stack Exchange OpenID. You can create a new one, but the Add Login page doesn't give you this option:
Fortunately, there's an easy (if non-obvious) work around.
Edit your profile, changing your contact email to the address for which you'd like to create a new OpenID:
Note: On the new profile pages, under Edit Profile & Settings, be sure to edit the email address in the Edit Profile sub-tab, not the one in the Preferences sub-tab.
Go to /users/account-recovery on the same site where you changed your email address in step #1 (this link goes to stackoverflow.com, so pay attention!), and enter the email you set in step #1.
When you receive the recovery email, there'll be a line with the text "If you'd like to use firstname.lastname@example.org to log in, click here to set a new password." Click the "click here" link:
Note: If you changed your email address on a different site from the one where you triggered account recovery, the email above will instead begin with the following text:
Someone requested an account recovery on Site Name Stack Exchange for email@example.com; however, that email doesn't match any active users on that site.
Don't worry! You can still follow the rest of the steps here.
You'll be prompted to enter a password, twice. This creates a new OpenID.
Then, you'll confirm you wish to add this OpenID to your account:
You're done! You can verify you've added the new login by clicking "My Logins" on the left side of your profile page's "Edit profile & Settings" section:
You can remove your old login here too, if you wish.