You could configure a User Event webhook to point to a service outside of Discourse that sends you a notification when a user makes any changes from on their profile page. A service like Zapier could be useful for handling the webhook: Discourse Webhooks and Zapier.
WIth the webhook approach, it would take some work to be able to determine that the user’s name property was what had been changed - you’d need to compare the webhook’s payload against user data that you’ve saved to an external database. If, instead, all you needed to know was that the user’s profile was updated, processing the webhook would be straightforward to process.