I found the problem why the "Remember my login" checkbox had stopped working on my site. I had secured the MediaWiki login page with SSL (https), and as a result, cookies were being created as secure, which made them inaccessible to the regular http site. Thus, I lost my session and got handed a new one.
To prevent this, I set $wgCookieSecure = false in LocalSettings.php, and all is well.
If anyone is interested in my secure login page "extension," here it is, ready for inclusion in LocalSettings.php:
http://meta.wikimedia.org/wiki/Help:Configuration_tips_and_tricks#HTTPS_ on_Login_only
- DanB (maiden_taiwan on meta.wikimedia.org)