What will she do, and who will be paying?
In my business, I deal with seniors either already using, or wishing to use the internet. In almost every case, the request has been the result of children's or grand children's recommendations.
Before spending money on equipment, associated connection gear and ongoing internet service fees, you should try to understand the motivation for the request, and whether there are more cost-effective alternatives. Consider also the technical background of the senior.
For example, a senior who spent his/her working years using computer equipment will feel more comfortable with a computer than someone who has no experience with computers or typing. You must take into account the person's dexterity (ability to comfortably use a keyboard and mouse, or even a touch screen), their eye sight (for determining size of screen) and their ability to pick up new skills.
If the goal is to give your grand mother a means of frequently communicating with grand children and getting to see them (via skype or other audio-visual applications), make sure you get equipment that will provide adequate bandwidth and compatibility with equipment on the "other end" of the line.
If the goal is to provide your grand mother with a form of entertainment (playing games, either alone or with online partners), you may be able to get away with a PC which already has games on it, and has the capacity for you to install other games via a CD or USB stick (with no internet connection required).
If the goal is to provide a means of viewing family photos in a convenient manner, you might consider getting a wifi-enabled picture frame. These will allow you to load and update her frame remotely through a special email address associated with her frame.
Regardless what you decide, understand that there will be ongoing costs, and probably requests to show grandma how to use whatever device she has. If you don't want to be responsible for the ongoing operating costs and training demands, you might want to rethink your strategy.
So now, some actual recommendations:
If you and your family are Apple-centric, and you insist on buying a device, get the iPad. It's the most intuitive of tablets, and despite the cost, you will probably feel most comfortable having her use a device you can understand.
If you and your family are PC-based, you might want to consider buying a Windows-based tablet or notebook PC. Just make sure it has a large enough form factor to offer a comfortable keyboard and screen.
Finally, if your grand mother already has experience using an Android-based cell phone and is an avid reader, she should be happy with any of a number of tablets, including the Amazon Kindle or Barnes and Noble Nook.