I had a the issue with Firefox 3 that the log in for several websites were not saved due to a failure with my cookies and I would have to relogin every time I restarted Firefox.
First, check the menu Tools>Options, select the Privacy tab, and make sure that "Accept cookies from sites" and "Ask me before clearing my private data" are checked and "Always clear my private data when I close Firefox" is not checked.
If that doesn't solve the issue, look for a file called cookies.sqlite in the profile directory, close Firefox, send the file to the recycle bin, open Firefox. Firefox will replace the file and everything should be working again.
On my computer, the cookies.sqlite was in the hidden directory "Application Data" and you will have to show hidden files and folders to see this folder. Here is the full directory address:
C:\Documents and Settings\<Username>\Application Data\Mozilla\Firefox\Profiles\<profile>.default\