I set up my child's phone with my Apple ID. Now all of the text messages come over with my name instead of her name. Can I change this on the iphone without setting up an apple id for her?
No. You have to set up an AppleID for her. If you want to share apps and music purchases, then you can use Family Sharing, and you can even specify if your child is able to purchase music/apps, or require you to approve the purchase first. This way, she is not able to rack up a huge bill, and you can both have separate messaging accounts.