It's easier than you think.
Show her an immediate, practical and continuing benefit. (this may help decide whether you should be using twitter too!)
Although Stephen Fry is everyones favourite grandson being able to follow his every movement is probably not sufficient justification for your Granny. Still hundreds of thousands of followers hang on his every tweet.
I'm looking forward to Stephen following Charles Dickens example and producing novels in instalments, but using twitter. One hundred and forty characters per chapter now there's a challenge!