I have Mail set up to use my Gmail account, which uses the correct name, and it's not an Apple Mail problem -- I tested by sending an email to myself from Apple Mail on my laptop. I think what's going on is that my mother (who bought the phone for me as a gift) registered it under the wrong name -- my name is Kay, but she insists on calling me Katie. Does anybody know how I can confirm this and change it to the correct name?
The name shown with sent email is based on the name entered in the name field for the email account settings with the email client used.
If you are unable to change this with the account settings on the iPhone, delete the account followed by recreating the account entering the name you want to use in the Name Field with the account settings.