Final answer:
The permission needed to set the running user of a dashboard to someone other than yourself is the Manage Dynamic Dashboards permission.
Step-by-step explanation:
The permission required to set the running user other than yourself in a Salesforce dashboard is C. Manage Dynamic Dashboards permission. The Manage Dynamic Dashboards permission enables you to set or change the running user of a dynamic dashboard. A dynamic dashboard runs using the security settings of the designated running user rather than the person viewing the dashboard, which makes it possible to control the data visibility for different users.
It's important to differentiate this from other permissions like View Dashboard which simply allows a user to see dashboards Edit Dashboard which allows a user to make changes to a dashboard and Customize Application, which permits users to customize the application in various ways including creating custom objects, fields and tabs as well as modifying page layouts and other application-wide settings.