I may have found a solution that I haven't seen suggested yet. Here's our background so that you can decide if your mileage may vary:
* We're running OS X 10.8 "Mountain Lion", but the password rejection problem started with OS X 10.7 "Lion".
* I can't say for sure when it started, but it feels like at least 3-4 weeks now.
* We have no password rejection problems with Yahoo! mail on three other devices (two iPhones and an iPad).
* We are longtime users of Yahoo! e-mail as part of our AT&T DSL service which we started waaaaay back when it was Pacific Bell DSL. So well over ten years now.
* So we have both "@yahoo.com" and "@pacbell.net" e-mail addresses, but all of them work with Yahoo!'s mail servers for client-based, web-based, and iDevice e-mail.
* We currently have Mail set up with three accounts on our MacBook Pro: two "@yahoo.com" accounts and one "@pacbell.net" account. Only one of the "@yahoo.com" accounts is having this problem.
* I realized today that what's "special" about the account that's having the problem is that it's the "main" account directly linked/associated with our AT&T DSL account.
MY SOLUTION: For the affected account, I changed the user name from "_____@yahoo.com" to the primary AT&T DSL user name without an @suffix of any kind. This is the AT&T DSL (formerly Pacific Bell DSL) user name that I originally (a long, long, long time ago) tethered to Yahoo!. It's hard to remember/describe exactly which user name this is, but it may help to say that this is also the same user name I can use without any "@suffix" to log into Yahoo! web mail.
I have quit and restarted Mail about five times now and haven't been prompted for a password. Time will tell if it's a permanent solution.