How are Abram and Lot related?
Genesis 11:31 - And Terah took Abram his son, and Lot the son of Haran his son's son, and Sarai his daughter in law, his son Abram's wife; and they went forth with them from Ur of the Chaldees, to go into the land of Canaan; and they came unto Haran, and dwelt there.
In Chapter 13 you see that Lot pitches his tent toward Soddom, which he later figures out was a bad plan because shortly there after they go to war. When word gets to Abram that Lot has been captured he has a weird response...he says his "brother" has been taken captive. You can see clearly from Genesis 11:31 that Lot is his brother Haran's son...making Lot his Nephew. Why does the KJV record that Lot is Abram's Brother?
There is no doubt that Lot is Abram's nephew, established in Genesis 11:31, however in Genesis 13:8 it says, "And Abram said unto Lot, Let there be no strife, I pray thee, between me and thee, and between my herdsmen and thy herdsmen; for we be brethren." Therefore Abram initiated the action to eliminate strife because they were brethren.
Upon hearing of Lots capture I believe Abram "armed his trained servants" not in the spirit of familial obligation but of spiritual brotherly love, and so indicated by referring to him as "brother" just like you or I would refer to another Christian as "sister" or "brother"
Also, Abram "believed in the LORD; and he counted it to him for righteousness." (Genesis 15:6) Abram had a heart soft toward God and as such would have responded to a truth later expressed in James 4:17 "Therefore to him that knoweth to do good, and doeth it not, to him it is sin." Rescuing Lot and his family from Sodom was right.
One last thought, Lot was Abram's nephew, brother and friend. Any one of these titles would be appropriate for Abram to use toward Lot depending on the circumstances.