Currently Aiven does not support renaming existing service. Service name can only be set when creating the service.
If you need to have your service running under a different name, currently the best option is to create a new service, and manually migrate the data.
For PostgreSQL, creating a database fork and pointing clients to the new database is the easiest way to handle data migration. Do note that after creating a fork, writes to the original database are not synced, so database writes should be stopped before forking.
Get in touch with Aiven support either over Intercom chat or with email@example.com email in case you have questions about the process.