My wife has about six different email accounts for her jobs, so I have some experience how frustrating this can be.
Off the top of my head, here are a couple ideas.
The SMTP servers might need authentication to be turned on, or they might have some other misconfiguration. Apple Mail hides the SMTP settings in its preferences, so working with the accounts in the Accounts section doesn't help.
The SMTP servers may be rate limited, and your friend's friend could be running into this. There was one particular server that would make my wife wait for five to ten minutes after sending an email before sending another.
Gmail with two-factor authentication has application specific passwords which are separate from the regular Gmail password, and they don't need to be changed.
More information about the servers would be helpful.
She's using AOL as her personal mail account, and she using a regular Gmail account, and one business account that somehow maps to the gmail account. She's a missionary at my mom's church, so I'm kinda loathe to name the organization.
And you're right, it wasn't in the account tab, it was one of the other tabs (Open Mail, click the Mail option at the top left, and it was one of those options). In one of those, it showed 4 smtp servers (Gmail, AOL, smtp.gmail.com, and smtp.aol.com) as all being "offline". There was an option to "take all accounts online", but it did nothing. We verified her passwords, and all the various SMTP acct options (port number, require SSL, etc...)
I ended up passing the buck off to her IT at the missionary group, since I have very little experience with Macs (I've used Amigas and Apple IIe's more than I have modern Macs). I was just curious, since I may run into the issue again with other folks. My mom has a terrible habit of trying to make me the volunteer IT guy for her church friends...
Lastly, one of my first jobs in IT was as admin for Exchange 5.5, and whatever mail server that Sun box ran, AND I HATED EVERY SECOND OF IT. I'd rather be stuck in the hot sun repairing fiber bundle cuts, or standing on top of a watertower, trying to aim a parabolic dish at it's mate 5 miles away using nothing but binoculars and a multimeter.