This article indicates that Umar I would refer to Umar ibn al-Khattab. It gives a tale of Umar ibn al-Khattab offering marriage to his son to a particularly honest woman who would eventually become Umar II grandmother. It then gives a brief biography of Umar II explaining why he is so revered to this day by Muslims regardless of sect.
While I was looking through a couple of other articles on the subject I also found out a few more interesting facts:
-Umar ibn al-Khattab once said that a descendant of his would be scarred and would right injustices during his rule. Umar ibn Abdul-Aziz was scarred on his face during his childhood by an animal and was widely seen as a just ruler.
-There was an Umayyad practice of cursing Ali’s name at the end of sermons. In an attempt to reconcile the Muslim factions, Umar ended this and replaced it with a verse from the Quran: “God commands you to practice justice, enjoins you to help and assist your kin and He forbids obscenity, evil or oppression, so that you may remember Him.”