http://badger.com is pretty awesome. Has a lot of pre-configured one-click "apps" which can be tremendously helpful. For example, click the "Tumblr" button and it automatically takes care of all the DNS re-arrangements required to serve up your tumblr on your domain.
Very slick and easy to use interface. I've had a lot of good experiences with their customer support too.
NameCheap.com - It's much more reliable than GoDaddy and doesn't try to upsell as much. Plus, the signup process is rather straight forward. I can help your grandma register a name if you'd like. Just have her email me at support(at)namemage.com and I'll be glad to help answer any questions she has when registering.
I second NameCheap. The sign up process is fairly straightforward and free of un-needed upsells, but may be confusing to people unfamiliar with basic internet jargon/terms. However, I wonder how "easy" sign up needs to be if the hypothetical mother in question is savvy enough to know that she wants to purchase a domain name, rather than say, register her vanity URL on Facebook and call it a day.
Thanks for the tip, I'll definitely keep it in mind. The upsells are a bit troubling to me, though. Mom doesn't need SSL and certainly no website creation tools or hosting (WordPress all the way). I'd like to recommend a registrar that really just focuses on domains.
I'd send my mom to a web hosting service that'd register it for me. Otherwise she'll get confused and wonder why she can't do anything with it afterwards.
Very slick and easy to use interface. I've had a lot of good experiences with their customer support too.