Yes, it is normal. It will be important for you not to take this as a personal rejection. Instead, just regard it as 'one of those thi9ngs that little kids do'. It is very normal.
Is this normal behavior? -- your words
Probably. I'm wondering if your granddaughter equates "going places" with her grandfather and "staying home" with her grandmother. One of our grandchildren was quite inflexible in this manner - when young, she was not able to transfer this information from one person to another. Actually, I think this is not an uncommon occurrence. Hope this helps ....
If it helps my grandson is the same. I look after him one day a week and he is very loviing towards me when no one else is around but as soon as someone else is there he changes and when I visit hinm at home he never wants to be near me only his grandad. What really hurts is that he appears only to be like that with me. It's hard not to take it personally and I feel sometimes I must be doing sotmehing really wrong but I am not sure what as I do not smother him and if he does not want to know me then I leave him alone, I never force the issue. I am told however this is normal and I really hope he grows out of it soon.