Snapchat undoubtedly allows you to change your display name. You can even change someone else's Snapchat display name, but there's no option for you to change your Snapchat username. Snapchat has this policy as a security measure against harassment.
However, if you don't like your username and want to get a new one without losing your contacts, there's a workaround. You'll just need to create an all new username before getting rid of your old username. Here's how to.
Step #1. Launch Snapchat on your iPhone → go to your profile by tapping on the Snapchat logo at the top of the screen.
Step #6. Tap on Add from Address Book and Click on Continue at the bottom of the screen.
Step #7. Allow Snapchat to access your contacts by tapping OK.
Step #8. In your address book, people who're using Snapchat will have a Snapchat username under their name instead of a phone number. You can add anyone to your address book.
Step #10. Tap on My Friends and write down the usernames of the people who have not been added to your new account.
Step #12. Log in to your new account and tap on Snapchat icon again.
Step #13. Manually add the rest of your friends by going to Add friends and tapping on Add by Username.
Step #14. Now you can delete your old Snapchat account.
There's an alternate option as well. Simply make a new account and tell your friends about it. That one would be pretty easy, I guess!