The Borg drone who gains individuality in “I, Borg” is Hugh. He is initially identified by the Borg designation Third of Five, but the Enterprise crew gives him the personal name Hugh as he begins developing an independent identity.
The Enterprise finds Hugh as the sole survivor of a crashed Borg scout vessel. After being separated from the Collective, he starts showing curiosity, emotion, personal preferences, and the ability to refer to himself as “I.” The crew initially considers implanting a destructive formula in him so it will spread through the Borg Collective, but Picard ultimately rejects the plan as morally equivalent to genocide.
Hugh chooses to return to the Borg because he fears they will pursue him and endanger the Enterprise. His individuality later affects other drones, becoming a major part of the follow-up episode “Descent,” where rogue Borg seek freedom from the Collective. A common mix-up is Seven of Nine, another famous liberated Borg, but she is introduced later in Star Trek: Voyager, not in “I, Borg.”