Thanks for the tip! Changing the user name to just "name" vs "name@domain.com" for the SMTP server worked for me. The most obvious symptom was that no email had been received since whatever it was happened, but it was the *outgoing* user name that had to be changed. (I don't think it had anything to do with it, but I did change the "Full Name" field from the user's full name to just their login name as part of all that I was trying. Changing it back to her correct full name didn't cause the problem to recur, so I'm guessing that it was just the SMTP user name that was the problem.)
The utterly bizarre thing was that the previous setup had been working OK for a number of days, and then suddenly, from one morning to the next afternoon, the computer was endlessly prompting for the password. It was working fine in the morning of March 31, 2016 US EDT, but as of the afternoon of April 1, it was constantly asking for the password. (Mentioning dates JIC someone can correlate this with a silent update of some sort that Apple might have done.) As far as we could tell, there hadn't been any automatic update installed overnight; although the machine was rebooted in the late afternoon of the 31st. (So maybe a previously-downloaded update got loaded then? The last update the App Store app shows as having been installed was iTunes 12.3.3, on March 25, which doesn't seem like it could have been related. Before that, the last update was Security Update 2016-001 for OS X version 10.9.5, just before the upgrade to El Cap.)
I also had to uncheck the box under Preferences/Accounts/Advanced that said "Automatically detect and maintain account settings" because it kept changing the port number to (a number I don't remember right now) from the needed 993. (Needed for our particular email provider LuxSci, that is.)
***? This is bizarre; how could it have been working just fine with all the settings as they were, then suddenly require that the SMTP user name be changed, from one day to the next?