Impact of changing server name SQL Server


I have an interesting subject at this moment, so I want to open a little debate. The subject is about SQL Server. We copied an server X to create a second Y mirror. We applied all possible configurations. But we also need to change the Server instance name.

The debate is about the possible consequent during the chanegement of the server name. What should we watch out for?

Namely: in our case, we have just one instance.