Answer:
North Carolina is the tarheel state. It is not a name that merely applies to the university. There have been several variations I have heard over the years, but this is the most consistent. If you have ever been to NC you know that there are an abundance of pine trees. During the days of colonialism and mercantilism, NC was important for its supply of naval stores (tar, pitch, and lumber). Naval stores and tar were often identified with NC (which was often overshadowed by its neighbors VA and SC). During the Revolutionary War (occasionally stories say civil war, but the most reliable sources say Revolutionary), it is said that the NC soldiers put tar on their heels so they had to hold their ground. Therefore they became the Tarheels.
The term then became applied to the state. Since the university was the first state-supported university in the nation, it became known as the UNC Tarheels.