Though the place name can be found in English as Haran, Charan or Charran, some say that it should not be confused with the personal name Haran, borne by Abram's brother, among others. The biblical placename can mean "parched". The personal name Haran supposedly means "mountaineer" which is odd in a land hundreds of miles from the nearest mounain.
Haran is closely identified with Harran, presently a village of Sanliurfa, Turkey. The earliest records of Harran are the Ebla tablets, said to date from around 2300 BCE. Local guides claim to be able identify ruins that are supposed to be the house where Abraham lived and was married. However, repeated archeological excavations date only back to the Middle Ages and have yielded insufficient discoveries about the site's pre-medieval history or of its patriarchal era.
In Islamic tradition, the place Haran is positively associated with Aran, i.e. Haran, the brother of Abram/Abraham, father of Lot. It is most likely that the city became known as Haran because the man of that name lived there. He remained behind when his father Terah went south to Ur with his brother, Nahor. According to the Bible book of Genesis, Haran and Nahor's brother, Abraham, along with Haran's son, Lot, moved south th Ur when Abraham was born. Nahor married Lot's sister and Abraham married Sarah before Abraham recieved the call from God to move to Canaan. They stopped at Harran en route, where their father, Terah, died.
In summary, there is no doubt that this is the general area where Abraham lived for part of his life. He was born in that city at the confluence of the Tigris and Euphrates Rivers but many of the non-Biblical details are likely to be legend and conjecture.
However, it all does sell a LOT of souvenirs.