Formal:
* Designation: This is the most direct and common noun form of "designate", referring to the act of designating something.
* Appointment: This implies a formal selection for a specific role or position.
* Nomination: This suggests a formal proposal for a role or position.
Informal:
* Choice: This is a general term for something that is selected or chosen.
* Selection: This implies a careful choice from a group of options.
* Pick: This is a more informal synonym for "choice".
Example sentences:
* Formal: The designation of the new CEO is expected next week.
* Formal: The appointment of the committee members was made by the board.
* Informal: The manager's choice for the project leader was surprising.
The best noun to use will depend on the specific meaning you want to convey.