I might get laughed at here, but here goes....
1) Find out what she already does with her spare time. Her hobbies and interests in particular.
2) See in what areas she wants to grow. One thing older people tend to do is become set in their ways. You need to show her how the internet will bring value to her current interests instead of turning her on to things that younger people find cool. It MIGHT alienate her from the internet if she sees too many 2girls1cup reaction videos (sorry, I couldn't resist!!) on YouTube.
3) Show her some sites that are small enough that she doesn't get overwhelmed with information. I think the overwhelming factor is something that turns seniors off.
Depending on your granny's sensitivities, I would recommend putting a "safe browsing" application on your computer. You tend to forget how easy it was to stumble onto a site with flashing banner advertisements for Viagra during the late 90s.
I think as a first application, Google Earth (as you said a no-brainer) would be the place to start. The level of control over her own experience she would have as far as how much information she would access depends on her level of comfort; and hopefully over the course of a few days this should increase such that she is more comfortable absorbing new information. The metaphor of exploring the world in a virtual sense might help ease the transition to exploring the internet directly through a browser. Also, the gliding effect is calming in the sense that it's not a sudden jump: there are nice transitions between flying from one place to another and zooming in and out. The one problem you might face, however, is convincing her to use Google Earth over and over again; one way to do this is to provide her with a "travel itinerary" such that she can visit famous places to which she has never been.
I would recommend familiarizing her with one application per month.