Just an update - my problem seems to have resolved itself, but I can't say why exactly.
I access Gmail from three devices - my laptop at home, my desktop at work, and my iPhone. I disabled the Gmail account on my laptop before I went to work the other day, and also disabled it on my iPhone; this was to prevent my devices from accessing Gmail too frequently and triggering CAPTCHA, which I had just unlocked.
With the other devices off, Gmail still initially gave me some trouble on my desktop (a G5 running Leopard) when I first opened Mail, but restarting Mail resolved that. Later in the day, I turned on Gmail on the iPhone with no problem. Later that night, once I knew my work computer was off, I enabled the Gmail account in Mail on the laptop at home, and it logged in just fine, and started to download all of the e-mails that were deleted when I took the advice that deleting the account from my laptop might be the solution (it wasn't, at least not the only solution). Since then, everything's been fine (fingers crossed).
The only thing that I can suggest if others are still having this general problem is to disable the Gmail account in Mail and on all devices, unlock CAPTCHA via Gmail's website (
https://www.google.com/accounts/UnlockCaptcha?), and use the web to check Gmail while you wait a day before starting to turn devices running Mail back on one by one.