An eyepatch was a small pad that was often worn to cover a lost or damaged eye.
In Doctor Julian Bashir's holoprogram Julian Bashir, Secret Agent, the character of Falcon wore a patch, with an image of a falcon's head on, covering his left eye. After a transporter accident in 2372, Chief Miles O'Brien assumed Falcon's identity, including eye patch. (DS9: "Our Man Bashir")
In 2373, during an away mission on the Takarian homeworld, Commander Chakotay and Lieutenant Tom Paris met a Takarian bard who wore an eyepatch. The bard, who often sang the Song of the Sages, wore it to elicit sympathy claiming he needed money in order to buy ointment for his eye. His rouse was discovered when Lieutenant Paris saw that his patch had changed places during a second meeting. (VOY: "False Profits")