<a href="http://badger.com" rel="nofollow">http://badger.com</a> 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.<p>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'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.