What's the complexity of renaming a column in MySql?
It can be an O(1) operation if only the name is changing, see:

https://dev.mysql.com/doc/refman/8.0/en/alter-table.html

and search for: 


  Renaming, Redefining, and Reordering Columns


That is in a section that explains that the operation only modifies table metadata.

Back in MySQL 5.5 the manual says it works there too, except for InnoDB tables.