ALTER INDEX RENAME TO
Traditional ALTER INDEX statement is able to rename the index.
Login as Owner
You can change the index name by the owner.
SQL> show user
USER is "HR"
SQL> ALTER INDEX emp_emp_id_pk RENAME TO emp_id_pk;
Index altered.
Login as Other User
This operation can be done by other user as long as you qualify the index with its schema name for the first one.
SQL> show user
USER is "SYSTEM"
SQL> ALTER INDEX hr.emp_emp_id_pk RENAME TO emp_id_pk;
Index altered.
Although you can recreate an index with different name, we usually don't do it, because big indexes may take a long time to complete the creation.