A subject near and dear to my heart!
I will be 74 on the 14th (now to be known as the 4Steve day) so I guess I qualify as a senior. I have been a nerd since the 80's, have had an iPod and iPhone since launch, so when I got my ipad2 last spring, it was up and running in 5 minutes.
My husband, on the other hand, can barely figure out his way around it, and I work with him every daynto help him learn. He has made great progress at age 77 but it does take patience.
I have 66 apps currently running but I have picked out some that could get your grandmother started.
In no particular order:
Kindle
Overdrive Media (if she has a library card). Fantastic way to read for free.
Her bank (if it has an app)
USA Today
CNN
Flipboard (for all the reasons others have given)
The Weather Channel HD (she can even watch approaching storms on radar)
Smurfs Village (for sure! There are dozens of others if she finds she likes that kind of game)
Words With Friends HD (Play it with her, she will love the connection with you and you will improve your word skills

Planetary (fantastic way to enjoy all your music)
Wallpapers HD and Cool Wallpapers HD (both free. She will spend hours looking at backgrounds to use. You will have to teach her about using Camera Roll though)
I hope these help. I think what you are doing for her is just wonderful!!!