Once you have accepted a character you will asked to provide a name for the character. In general, the actual choice of a name is not important, but do keep in mind that it may have some effect on the game itself. For example, on some machines, the character name determines the filename that will be used to save the character to disk. The character name is used on the high score list.
You can play a dynasty of characters. If you use a Roman numeral at the end of your character name (like “Fred I” or “Pimplesnarg XVI”), the game will automatically increment the numeral each time you die (or win!).