Clear cache, cookies, and authenticated sessions - then restart the browser.There are known problems, though, with thoughtlessly clearing private data. Sometimes, it would be better to clear only the specific data that might be causing the problem.
It is possible to take a more detailed approach to clearing cookies, in many browsers. Here you should remember the various domains used by Blogger / Google, and consider possible problems caused by incorrectly removing some cookies, but not all.
In Chrome, Options - "Under The Hood" - Privacy provides the "Content Settings" wizard. The "All cookies and site data" wizard provides an inventory of which sites have stored private data. To remove cookies for a given site, highlight that site, then hit the "X" at the right.
In Firefox, the Tools - Options - Privacy wizard has a link to "remove individual cookies". To remove cookies for a given site, highlight that site, then hit the "Remove Cookies" button.
In Internet Explorer, the Internet Properties - "Browsing history", "Settings" gives you the "Temporary Internet Files and History Settings" wizard. Hitting the "View files" button opens another instance of Windows Explorer, positioned in "Temporary Internet Files". To remove cookies for a given site, simply locate and delete the specific files.
In Safari, the Preferences - Security wizard has a "Show Cookies" button. To remove cookies for a given site, highlight that site, then hit the "Remove" button.
In Opera, the Tools - Preferences - Advanced wizard has a Cookies menu. The "Manage Cookies" button provides the "Cookie Manager" wizard. To remove cookies for a given site, highlight that site, then hit the Delete button.
Selective removal of cookies will let you correct layered security problems that involve Blogger / Google, but possibly allow you to continue using other applications and sites without interruption. Just be careful, and remove exactly what needs to be removed - no more, and no less. And, as always, remember to restart the browser after removing cache and cookies.