what is this behavior called does it have a name?

Triangulation is the name for when someone 'brings in' a third person to sideline their partner or other close person.

It can be by taking the advice of the third party when it is identical to what you have been suggesting. Or buying something they rejected when you said you liked it, because the other person said it. Or mentioning that the other person dislikes something you've just said you do like. There are many ways to do it, but all are deniable and make the partner look jealous or unreasonable.

It is something that a lot of narcissists will do, but I don't think it is exclusive to narcissists. It is also passive aggressive, as it can be denied, and can be done subtly, so any intent to cause hurt is ambiguous.

I don't think any of that necessarily applies to the OP's sister though. I agree that she is likely trying to deny that people she loves are ill.

I agree with above posts.
However, I can see how OP can wonder why these behaviours need to be named.

My ex used to baffle me by forgetting the birthdays of people closest to him, whilst buying flashy presents and being very generous to people who are little more than strangers.

My son has also inherited this characteristic and I just don't get it. It isn't as if these people are even interested and are even sometimes downright embarrassed by it, as even they sense how inappropriate is it.

If anyone else recognises this trait in a family member I would also love to know it isn't just me!

Agree. Your sister is using denial. It’s a common protective mechanism, when in place it’s not there to be hurtful to others but to protect the individual from overwhelming feelings. Although I know it’s hard to be with gigi, it might help to think your sister has very strong reactions she can’t cope with, she’s not minimising what close family members are going through. Rather the opposite.

I was going to say exactly what AuntieE has said.
Your sister can only handle the stress and worry of serious and life-threatening illness among those closest to her by pretending it isn't happening and carrying on as normal. If it has a name, it's called being in denial.

I think you sister is just not able to cope with serious illness in the family and behaves like this because she is scared.

She can manage to discuss illness with those who do not mean so much to her, but not with those she loves.