I suggest you look at your anti virus program.... they have a nasty habit of caching passwords and inserting themselves in the transaction as a proxy. When it goes wrong in goes spectacularly wrong and very silently as far as the AV product is concerned.

Thank you for replying. That's a great thought, but I did find the answer, and it was not that.

Here's the rest of the story . . .

There was a webpage that the error notification referred me to, and I tried each suggestion there to no avail--except for setting gmail to allow unsafe applications. After all, Thunderbird on this laptop could log in with no problem before I changed to a more secure password, and Thunderbird on another laptop was having no problem. I googled the problem and searched Mozilla help in vain. Finally, in desperation I tried setting gmail to allow access to unsafe applications: this allowed Thunderbird to log in. (Both Thunderbirds are the latest: 38.6.0 on Windows 10.)

This led me to a technote on Mozilla's support site that mentioned that gmail does not consider Thunderbird version 38 to be a secure application because it doesn't use the Oauth2 security protocol. This, in turn, led me to discover in my Thunderbird instances that Oauth2 is an available security option (contrary to what the Mozilla technote implied). I then found that the Thunderbird that hadn't had the problem was already set to Oauth2, while the Thunderbird that couldn't log in was set to Normal Password. Once I set both Thunderbird instances to Oauth2, I was able to turn off access to insecure applications in gmail.

.....