Account missing
In "Options," "Saved Passwords," one of my Gmail accounts isn't showing up all of a sudden. (I have several accounts, and all ARE listed in "Account Settings.") Now every time I want to check this account's email, T'bird asks for my password. How do I get it to recognize this account and save its password? This has never happened before, and I've had this account for years.
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You'll need to tell Thunderbird to remember the password. There's a checkbox for this at the password prompt.
The problem is, now the Gmail account is showing up in "Options" as well as the password...but still, T'bird asks for it every time, even though I check the box for T'bird to remember it!
At the top right of the Thunderbird window, click the menu button 
> Options > Privacy & Security > Saved Passwords
If there's an old password for the account Thunderbird has remembered, delete it, and try again.