Try sympathizing with Alayna when she's feeling hurt, upset, angry. she doesn't need to be convinced that she shouldn't feel this way. She needs to know it's OK to feel this way. Tell her you know she feels like she has no friends. Don't try to convince her she's wrong. At the same time realize that she is over reacting and that this too shall pass. Once we accept our emotions, allow ourselves to feel them and grieve if it's a loss, as with a fight with a friend, or be angry when we feel mistreated, we are eventually able to let go of the feelings and move on. But when people are telling us we're over reacting, should get over it, shouldn't even be feeling so sensitive then we focus on defending those feelings and usually add more emotion because these people don't understand.
Most of us were taught that we shouldn't feel certain ways or that we should snap out of our feelings and we pass that belief on to our children by denying them their right to those feelings. This results in not learning how to accept and deal with how we feel.
I also strongly urge your sister to work with the therapist. No one, especially a child, lives in a world separate from others. How Alayna feels is directly related to what she's learned from others as well as how she feels about herself. She now needs to learn how to relate in a more satisfactory way to others and life events. Her family is part of the others. Her family also needs to learn how to act with Alayna. to help her change her world view. In family therapy, Alayna and her family will learn new skills and new ways of relating with each other.
What you've described as Alayna's behavior sounds very normal. All children have on again, off again relationships. All children say or do things to upset friends. They fight. Children are grossly unskilled in relationships. They have to learn how to deal with the ups and downs in life. The family is the most important place to learn this.
Did your sister not see the humor in the two girls yelling at each other? If not, I suggest that she takes a very serious view of things which makes it natural for her daughter to also be overly sensitive. Perhaps counseling would help both of them to take this sort of happening less seriously. Humor lightens the serious parts of life so that we can see them in perspective. Having a fight with a friend as something not so important and which will happen over and over in our lives. How we think and act greatly influences our happiness.
The two girls yelling at each other is very normal and to be expected. It really is no big deal except that it feels like a big deal to the girls with whom it's happening. As adults we can laugh because we've been there and know it's no big deal. At the same time we need to realize it is a big deal for the girls and sympathize with them while reassuring them that their feelings are normal and will pass with time. Hold her,. Tell her you know it hurts. It's good to feel the pain. Then when the crying stops tell her about your own experiences with fights and how next week they'll be friends again. Alayna may say, no she hates the other girl. Tell her that's OK too.
When the parent has confidence that this is a normal part of learning how to get along and can be accepting of the feelings and offer reassurance the reaction will pass more quickly. It's very important for this to not be a big deal for the parent. When the parent is anxious or upset the child will continue to be upset.
Perhaps your sister has had difficulties with friendships and fears that her daughter will too. Counseling will help her become more confident and help her to learn some skills that will help her help her daughter. Whatever she's doing now isn't working. It's not just a matter of the counselor changing the way Alayna thinks and reacts. It's also a matter of changing the way everyone around her is affecting her and she them.
You have good questions about how you can help her at home. The therapist can answer those questions. Answering questions and teaching skills is what therapists do. That is the reason for family therapy.
I suggest that your sister is also very sensitive and feels threatened at the idea of family therapy. Reassure her that the therapist is going to teach her new ways of looking at things which will increase her self-confidence. Then she'll be better skilled at helping her daughter. The therapist knows that she's doing the best that she can and that she is a good person and a good mother. All of us have sensitivities and fears and when we learn to overcome them we find that we're happier. It does take a leap of faith, an acceptance of our fear, and the knowledge that with help we can be a happier person. For her she'll learn how to help her daughter.
If she prefers she can focus on helping her daughter in family therapy. She doesn't have to be in therapy for herself, though I suggest that once she's able to relax and accept help for herself, too, she'll feel relief. I suggest that if she'll try out family therapy, with the idea it's to help Alayna and learn skills to help her she'll find that it's not so threatening and actually helpful.
Later: My 7yo grandson often refuses to try to do new things. If he can't do it the first time, he gives up and feel hopeless. There are ways to talk with children that can help them overcome their feelings of helplessness. I recommend Positive Discipline by Jane Nelson. It has a two page chart that lists the behavior, what the child is thinking/feeling, how it makes the parent think/feel and how to respond. The other resource is How To Talk so Children Will Listen & Listen So Kids will Talk by Adele Faber and Elaine Mazlish.
This therapist may not be a good fit for your sister and daughter. A student is less skilled and may not be as helpful as someone more experienced. It would be reasonable to ask for a different therapist to see if she's more comfortable with the therapist. At the same time it's important to be involved in family therapy. Another therapist is going to want to do family therapy, too.