The story of the name, from the sources
Omar comes from umr, life. Ibn Faris says the root means lasting, and the stretch of time. In Lisan al-Arab, al-umr is life, and from it comes the old Arab oath, la'amruk: by your life.
“a root that means lasting, and the stretch of time”
al-umr is life; hence the old oath la‘amruk, “by your life”
Maqayis al-Lugha · vol. 4, p. 140 · Lisan al-Arab · vol. 4, p. 601
And a small wonder of the language: the name Amr is written with an extra letter, a waw, and al-Jawhari explains it is there to tell it apart from Umar. In the Prophet's time, Amr was the common name: we counted the entries of al-Isaba, about one hundred and eighty named Amr, and about twenty-five named Umar.
عَمْرو · عُمَر
“‘Amr is written with a waw to tell it apart from ‘Umar”
In al-Isaba: about 180 named ‘Amr, about 25 named ‘Umar
al-Sihah · vol. 2, p. 756 · Taj al-Arus · vol. 13, p. 123 · Ismuka count in al-Isaba
Its most famous bearer is Umar ibn al-Khattab, may God be pleased with him. Ibn Sa'd reports that the Companions said: we are the believers, and Umar is our commander. So he was the first to be called Commander of the Faithful, and the first to date the years from the Hijra.
“We are the believers, and Umar is our commander”
the first “Commander of the Faithful”
and the first to date the years from the Hijra, in 16 AH
Ibn Sa’d, al-Tabaqat · vol. 3, p. 213
And the title al-Faruq? Al-Zuhri said: we heard that the People of the Book were the first to call Umar by it.
الفاروق
“We heard that the People of the Book were the first to call Umar al-Faruq”
al-Zuhri
Ibn Sa’d, al-Tabaqat · vol. 3, p. 205