Names can be given to hatchlings and Adults. They can be anything you want as long as they are clean or else your hatchling may die.
How to Name
The name action is found on your dragon's page where it says actions. Then you type a name in. No dragon can be named the same thing. For example, if you want to name your dragon "Sparky" and there is already a dragon named Sparky, it will not be named Sparky and 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 (Such as æ,ã,à,á,â,ä,å,ā,é,é,ë, ę,è,ê,ė,ē,ÿ,ü,ú,ū, û,ù,į,ì,ï,ī,î,í, œ,ō,ø,õ,ó,ö,ô,ò,ś,ß, š,ł,ź,ž,ż,ć,ç,č,ń,ñ, Japanese, Chinese, Arabic, Greek, etc alphabets), 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." 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 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.", meaning that your dragon has been named successfully.