In 2007, Google formed an integration deal with GoDaddy to enable GoDaddy hosting users to easily verify their sites with Google Webmaster Tools.
Sometime recently that verification process broke.
A Google Webmaster Help thread has one GoDaddy user complaining about this issue. He described it as follows:
As of today, when I click that same "Launch Google Wembaster Tools" icon, it prompts me to login. It seems Google has changed the way they accept parameters from Godaddy (or stopped accepting parameters from Godaddy). Now I can't access the nearly 10 domain webmaster tool dialogues, which are crucial to the maintenance of my sites.
Googler, Susan Moskwa explained they are aware of the issue and is working with GoDaddy to have it resolved. She said:
We're talking with GoDaddy about an issue that sounds like yours, so sit tight.
Alternatively, you could verify your site in another Webmaster Tools account (e.g. with the Google Account you use to post in this forum); sites can be verified in multiple accounts, and you'd have access to all the same data.
Forum discussion at Google Webmaster Help.