How do I gently correct my mother-in-law, who has had brain surgery, when she calls my daughters the wrong name?

Speaking as someone who has dealt with people with brain tumors and head injuries, before I'd try to correct your mother-in-law, I'd find out from her doctor (or other person who knows her case) whether she is capable of remembering names and accessing that memory for speech at the appropriate times. IF she is able to do it, I'd set up a time when I could be alone with her. Keeping in mind I'm no counselor, I'd then address it from the point of view that, "There is something I want to talk to you about, and I want you to know that I'd want you to talk to me about this if it ever happened to me. (etc.) I'm not sure if you've noticed, but sometimes you say Jill's name when you're talking to Jackie. And sometimes you say Jackie's name when you're talking to Jill." Saying 'sometimes' gives Mom a break from feeling like her problem is huge, and takes away from potential defensiveness, like from saying 'always'. So her response would not be "No, I don't!" Remember to start by saying "I love you" or your equivalent of that. You can also address it from the point of view of her being an adult, and how it affects your child. HOWEVER, if you find that the brain surgery has left her with some deficits, so that it is unreasonable to expect her to use correct names, etc., THAT will just be the case, and you and your daughters will have to find a way to deal with it. It's really amazing how children have the capacity for understanding and 'rising to the occasion', almost no matter the age of the child. It would then take a private discussion (age-appropriate) with the child for them to understand how the place in Grandma's brain, where she remembers your name and face, has been damaged. We hope she might get back the ability to call you by your correct name, but she might not. Just know, she knows who you are, and it's just finding and getting the correct name out that is difficult. She loves you even if she says the wrong name for you. Empathy is what this is about. Put yourself in her shoes. How would you want someone to address this issue with you? That usually helps. And if it's a permanent problem, realize you may have some grief about this and related losses. Let yourself feel, emote, in safe places and with loving people.