Hi
It is not presently possible to edit the watch's name as listed within the Health app.
You could unpair your Apple Watch and set it up again as a new watch. This would add it to the list of Data Sources with whatever name it is set it up with. However, the old watch name would also remain listed within the Health app.
If you are happy to lose all data stored by the Health app on your iPhone (including Activity data, Achievements and Workout data) - and also to lose other sensitive data on your iPhone including HomeKit data, saved passwords and Wi-Fi settings - then you can:
To reiterate: this would result in the permanent loss of all accumulated Activity and Health data from all sources.
You would also need to add any cards back to Apple Pay on both devices.
More information:
https://www.apple.com/uk/support/iphone/backup/
About encrypted backups in iTunes - Apple Support