As per my search, there doesn't appear to be any feature in the documentation that explicitly states you can change the name of a flow run once it has already started. The flow run name can be set dynamically at the time of flow creation using a function or a formatted string, but modifying it post-start isn't documented.
You may find more information on setting the flow run name dynamically in the documentation
here and on using the runtime context
here.
For specific use-cases or functionality alterations, it might be helpful to check GitHub issues or discussions in case others in the community have faced similar challenges. Would you like me to search GitHub for any related issues?