Complaints in the Blogger help forums about "Blogger won't remember me" are becoming pretty common. My general advice is to clear cache and cookies, and start over.
Maybe we're seeing a way out of the fog.
Joel, aka Far Outliers in Clicking "Remember Me" is Useless, may have found a clue.
At work, I was getting redirected successfully, but at home I wasn't--that is, until right now, when I unchecked the box in Firefox that allows 3rd-party cookies.
It makes sense that third party cookies would be part of a migration.
Cookies created by one server ("beta.blogger.com"), and read by another ("www2.blogger.com") might look like a third party relationship, to the browser.
Now, it's up for someone to verify this. If you're blocking third party cookies, and you haven't yet successfully gotten Blogger to recognise your login state from browser session to browser session, check your third party cookies filtering in your browser.
And if you find a setting that, when corrected lets you login properly, let us know.
For more insight into the cookie issue, see Roberto's Report: Another Cookie, anyone?.
(Edit 1/3): Besides the third party cookies problem, there's an interesting possibility to explain first party cookies, and how they could present a problem in both Firefox and Internet Explorer.