1. Activate Siri by holding down the home button and wait for the beep.
2. Say "Call me ____" Fill in the blank with whatever you want Siri to refer to you as. This can be your name or anything else.
3. Siri will reply "From now on, I'll call you ____. OK?
4. Reply "OK."
Siri will the update the nickname field of your contact info to this name. This is also how your name will appear when Siri types out your name.