Unable to add new OpenID identity in Drupal 7
Thanks to Mołot's comment, I was able to figure it out.  The problem was the administrative overlay.

Using the Chrome Developer Tools console (Chrome Menu/View/Developer/Developer Tools) I was able to see the request history.  The last request included:

referer:http://mysite.com/user/1/openid?render=overlay&amp;render=overlay


The response from Google also contained a header that looks like an error, although it may not have been relevant:

p3p:CP="This is not a P3P policy! See http://www.google.com/support/accounts/bin/answer.py?hl=en&amp;answer=151657 for more info."


I don't know if the referrer header was the actual problem, or if it just reflected a problem caused by including "render=overlay" in the request going on in the background between my server and the OpenID provider.  In any case, by turning the administrative overlay off (Admin/Modules/Core/Overlay), I was able to add and use my new OpenID.  It worked for both Google and Yahoo.

I would also add, as noted elsewhere, the URL to use to set up the OpenID for Google is:

http://www.google.com/accounts/o8/id


That initially confused me because that does not appear to be a unique ID.  However, using that ID both to add the OpenID for Drupal, and to log in using OpenID, works.  The string stored as my "OpenID" is different, and has what appears to be a unique identifier, but the user never has to enter that long, unique string.

Not only did the Administrative Overlay prevent the OpenID from being added, it also prevented me from seeing the error message when I used the wrong string to add the OpenID.