Part of growing up is to be able to overcome adversity by herself. When someone is unhappy and wants to make a change, the question is how will the new environment make one happier. What is it about the school she is at that’s making her so unhappy? From your brief post, you said she doesn’t have many friends. How is changing school going to make it better?
My college senior daughter had a very difficult year due to change in our family, BF breakup and not certain what she would do after graduation. I listened to her cry often on the phone. It is gut wrenching when you hear your kid cry. At the same time, I knew she had to figure it out by herself. I got her a counselor, which she thought was very helpful. She is doing much better now. She came up with a game plan for next year. She just had her best semester ever with 4.15 GPA. I think what’s more important is knowing she could work through adversity by herself is giving her more confidence.
I understand OP’s daughter is a lot younger than my kid and probably needs more of OP’s help to work this through. I would have her stick it out until end of year instead of running away. She can start sending out applications to other schools and make her decision over the summer. This is no different than when we don’t like our jobs. We get a new job before we quit. Getting 3.9 the first semester at a top tier school means she is spending too much time studying instead of having fun. Has she joined any club? What about a job on campus? Both of my kids worked while in school. The younger one works in an office that gives out funding for various community projects. It has given her an opportunity to work with many professors and student leaders. They also had singles freshman year, and they loved their privacy. Tell your daughter to leave her door open when doesn’t need privacy, have some food (like fresh popcorn) in her room to attract people to visit. Try not to study her room, instead, go do her work at a more social library (my kids knew which library to go to in order to see and be seen). Encourage your daughter to see a counselor on campus.
Instead of “allowing” your kid to pack up and come home whenever she is unhappy, this maybe a teaching moment for her to learn how to cope and come out on the other side. This is hard because no parent wants to see our kids unhappy. Good luck.