You can designate any one character your Star Character from that character's Inventory screen; scroll to the bottom and there is a Star this Character button. Your starred character gains increased stats (+100 HP, +250 ATK), appears in your profile with a custom message when viewed by other players, and a Star icon appears in the top right next to the character's name when viewing the character's status info. You can also choose to Tweet your star character when you perform a Twitter Restore. After setting a star character, you must wait 24 hours to set a different star character.
There is no downside to starring a character, aside from the 24-hour cooldown, so make sure someone is always starred!