Only certain Races allow runners to transfer their registration to a different event/participant on RunSignUp.
If transfers are enabled for the race you have signed up for, then you will find the option to do so by going to your Profile -> My Registered Races -> Manage Registration. There are two different types of transfers ("Event to Event" and "Participant to Participant").
If transfers are not enabled, then you will need to contact the Race directly to learn more about their transfer policy. You can contact the race directly by going to the Race Contact Infosection of the race page and filling out the Questions form, or, if you are already registered, you can use the email listed at the bottom of your registration confirmation email.