Possessive of a proper name

Eh... this is a weird area. I'd say it's incorrect, but that's because I follow certain style guides. Since my last name is Davis, this is a matter of more than hypothetical interest to me. I would say "Barbara Davis' post on this thread" is incorrect if someone wrote it about my post. But if Charles wrote that Charles' post was worth reading, I wouldn't correct him on his own name. However, if you hire me as your proofreader, I will "correct" it every time.

(I'm not too worried about giving away my identity - it's the sixth most common name in the U.S. Even in my tiny town, three people with my name have a library card and four have memberships at the YMCA.)

Yes, the AP Style Book should be the ruling authority on this. My copy is at work and I am home. Although I have to say, the AP style on this is really odd, which is why my company decided to not follow it for this rule (except on press releases where AP is expected). Repost to this on Monday when I'm back in the office to remind me, and I'll look it up and quote it for you.

I would, without hesitation, write it as Eilis's mom.

One of the the other factors to consider (as you have written) is how it's said.

Richards' apologies is said "richards apology" and Eilis's mom is said "eilishiz mom."

Eilis' mom reads like it should be "eye-lish mom" -- and that's just not what's said. No matter how awesome and special she is, she's not Socrates or Moses.