Hello Hollch,
Thank you for reaching out in Apple Support Communities. We understand you'd like to change the name for your daughter's Apple Watch for the Apple Watch face in Schooltime. We can help point you in the right direction.
We recommend to open the Watch app on the paired iPhone and tap the My Watch tab in the bottom left corner and go to General > About > Name. Then, you can re-name your daughter's Apple Watch. Here's an article with that information: Change the name of your Apple Watch - Apple Support
Performing this step can help update the name for the Apple Watch.
Kind regards.