Names can be given to hatchlings, frozen hatchlings and adults. They can be anything you want as long as they are clean or else your dragon may be killed. Other consequences can also be put into action, such as loss of the naming ability, may also be put into play if it is serious enough.
How to Name
The name action is found on your dragon's page, underneath the 'actions' link. After clicking 'name,' you type a name in. No dragon can be named the same thing. If you want to name your dragon a certain name, and there is already a dragon named that, it will not be named and the message "You try to write the name, but the ink disappears as you write, so you decide to try something else." will appear. If you try to give the dragon a name involving characters from an alphabet other than English (For example, mathematical symbols,) you will get the message, "You try to write the name, but some characters disappear as you write, so you decide to try something else." This will also show if you put a space at the end of your dragon's name.
Also, if you type in a name and wait a few seconds before pressing the "Name" button, a small X or check mark will appear beside the name, telling you whether it is acceptable or not. If the name is too long you will get the message "You realize there's not enough room to fit that name, so you decide to try something else." If you do not wait and click on "Name", either of the messages from above will appear, or you will get "You write the name next to your dragon's picture in the scroll." This message recognizes that the naming is successful and your dragon will be named so.