I totally feel your pain because I get this all the time myself. Kids today are not as good at commitment as they used to be and they stink at prioritizing things.
I would say let him go to see grandma. She has more years behind her than she does in front of her - even if she lives to be 100. I don't think you would look weak or backtracking if you let him go either. This could be considered a family emergency because of the other family member having the trouble (can't remember what it was).
I don't think you can really argue with family commitments. I tell my guys only three things are more important than baseball - family, church and school. Now if family, church and school keep causing someone to miss then we need to sit down and talk about whether baseball is important enough to them. If this other stuff gets in the way they need to give up something. Maybe even baseball.
A couple of years ago we pulled two upsets and won our district tournament which put us into the regional tournament the next week. After the first game in the district tournament this kid comes up to me and said he was going to miss the rest of the week (meaning district championship and practice leading up to regionals) to go to the beach. I asked him if his parents would let him stay with another family member so he could stay. He tells me he wasn't going with his family but a buddy of his. I was dumbstruck. I looked at him and said you make the decision you think is best for you - go on vacation or stay with your team. Needless to say he went on vacation.
We are practicing on Sunday for our game in regional on Monday. Some of the guys asked me about this kid because he called them from the beach to find out what I was going to do. I told them he was not allowed back. He calls me late Sunday night crying saying I was kicking off the team. He could not understand he quit the team.
My point is I don't think I can tell a family when to take vacation. Surely / hopefully they are smart enough to play around the season but sometimes the job won't allow for that. If this kid had went with his family I would not have been happy but I can't stop it.
Now I do control playing time so if this does happen Little Johnny might not be playing for sometime or he is doing some running when he comes back.
Let this kid go and sit him for a game or have him do some extra running. Explain it to the team because more than likely they will understand.
It's not perfect but it pretty much works for me.