While the answer is extremely simple, time to time it takes too much time to figure it out what you need to get it done.

Here is a checklist of what you need to make it happen:

• Joomla! version.  It should be Joomla! 1.5.20 Stable [ senu takaa ] 18-July-2010 18:00 GMT

• PHP version. It should be PHP5, BUT not 5.3 OpenId PHP library included in Joomla! OpenID plugin is somehow incompatible with PHP 5.3

• OpenID plugin should be enabled:

• We found it usefull to change Name/username parameter value to "Name" in the mod_login parameters:

That's it! Go and try it.

P.S.

I have heard a rumors and read this discussion: http://groups.google.com/group/joomlabugsquad/browse_thread/thread/adeffbc94629e332 and i am going to be dissappointed. If Joomla! team will really exclude OpenId from the system in 1.6 release,  and this featur become unavailable for my sites, then it will be a good reason for me to jump to one of the alternative CMS.

P.P.S. Joomla! Bug Squad!!!! Please do not do this!!!!