Luna's fear could be of being left completely alone. She accepted her mother's death, but still had her father. I don't know if she'd be able to come to terms with that quite as well. Also, after her fourth year she has friends like she hasn't had before. Judging by how her bedroom was decorated the idea of losing them is probably quite frightening as well. She may not need people around her all the time, but finding herself with no one could be a fear of hers.