How about teaching her to cook? It is a skill she is going to need for the rest of her life, after all.
Start with whatever she likes best to eat. Let her do the shopping for whatever you are making, with a list provided by you and a suitable amount of money for what she is to buy.
Tell her honestly that you are afraid she will be bored staying so far away from her friends. I gather there has been an upset in her immediate family - she is old enough to understand a good deal of what is going on, but not all of it.
Have you told her that you are delighted to have her for the school holidays, but sad (cross? vexed?) about the reason for it? Does she know she can talk honestly to you about it, if she wants, and that you won't repeat anything she tells you in confidence?
If you haven't told her this, please do so. She is at an age where girls, particularly, want to be considered grown-up. You and I know she isn't anywhere near grown-up now, but she is old enough to understand some of the ramifications of what is going on in the family right now AND to be worrying about it.
Ask her too if she likes visiting her great-grandmother with you, or would rather not.
You didn't mention whether you have a husband or not, but if her grandfather is around, then she and he could do something while you visit your mum.
Oh, and brutally practical, but I have had some nasty surprises teaching girls her age, are you sure her mother had explained periods to her? Believe it or not, I have had girls of eleven and twelve crying in shock at the sight of their first period and totally unprepared for it, so a chat about what our mothers called " developing" or "the facts of life" might just be a very good idea.