The word Hebrew comes from a root word meaning to stand on the other side. Some scholars think that the term was applied to Abram because he crossed the River Euphrates to come to the land of Canaan.
Joshua's review of Israel's history would seem to support this idea as he mentions crossing rivers several times:
3 I took your father Abraham from beyond the River, and led him throughout all the land of Canaan, and multiplied his offspring, and gave him Isaac.
11 “ ‘You went over the Jordan, and came to Jericho. The men of Jericho fought against you, the Amorite, the Perizzite, the Canaanite, the Hittite, the Girgashite, the Hivite, and the Jebusite; and I delivered them into your hand.
15 If it seems evil to you to serve the LORD, choose today whom you will serve; whether the gods which your fathers served that were beyond the River, or the gods of the Amorites, in whose land you dwell; but as for me and my house, we will serve the LORD.”
Other scholars think the name comes from Eber one of Abraham's ancestors mentioned in Genesis:
24 Arpachshad became the father of Shelah. Shelah became the father of Eber. 25 To Eber were born two sons. The name of the one was Peleg, for in his days the earth was divided. His brother’s name was Joktan.