Clan cats have very important rules for how they are named. The mother chooses the first name, and the leader chooses the rest. See Ceremony Templates. Have fun!

When a kit is born, their mother chooses the first part of their name based on either physical attributes, or personality. Usually names based on personality are given to the kit as it grows a little more. The name must be something that wild cats know of; other wild animals, herbs, character traits, parts of nature, etc.
Example: Rose
The name Rose might have been given to the kit because it is very beautiful, like a rose, or it's very loving, because roses are often associated with hearts and love.

For kits, their mother would name them, and the end of their name would automatically be 'kit' until it became an apprentice.
Example: Rosekit

As an apprentice, the end of the name would change into 'paw' but the beginning would stay the same. Once again, this changes as the rank changes.
Example: Rosepaw

When apprentices become warriors, their leader chooses the end of their name. They keep the first part, but the end can be almost anything, as long as wild cats are aware of it. The most common name endings are claw, face, pad, foot, belly, tail, eye, ear, heart, storm, cloud, etc.
Example: Rosetail

Sometimes an apprentice is given a warrior name with a name ending that is different from those. The ending is usually revelant to the beginning when this happens.
Example: Rosethorn

Once they earn their warrior name, they keep it for the rest of their life, except for in these cases:

a) They become the leader of their Clan, and they earn their leader name from StarClan. All leader's names end in 'star.'
Example: Rosestar

b) They become an elder in their Clan, or they get a major injury. In this case you would use a name-change ceremony (see Ceremony Templates) and change their name.
Example: Missingfoot
(Maybe Rosetail/Rosethorn was in a battle and lost a leg, and when she became an elder, she changed her name)