The easy and safe way

Open Terminal with Ctrl+Alt+T, and type:

sudo chfn username

where username is what shows in front of the @ in the prompt.

Your existing "display name" (shown at the login screen, on the top right of the desktop, etc.) will appear enclosed in brackets [] after Full Name:. Type in whatever you want the new "display name" to be and press enter.

You can leave the remaining fields (Room number, Phone, etc.) blank (just keep pressing enter).

Once done, exit the terminal and logout. You should have a shiny new "display name"!

The direct but risky way

The original version of this answer proposed directly editing /etc/passwd to change the "display name". This is risky if you don't know the format of the passwd file, since a typo could leave you unable to login any more. You have been warned.

The general format is:

username:x:1000:1000:displayname,,,,:/home/username:/bin/bash

Change "displayname" to whatever you want the name displayed on the login screen to be, e.g. "John Doe" or just the same as "username". This name CANNOT contain colons, commas or equal-to signs.

Save and quit your editor, and PLEASE then use pwck to check that the format of /etc/passwd is still valid, using:

sudo pwck | egrep -v "directory.*does not exist"

The response you are looking for is:

pwck: no changes

Otherwise, just logout, and you should see your chosen display name.