Umar II was an Umayyad caliph who ruled from 717 to 720. He was also a cousin of the former caliph, being the son of Abd al-Malik's younger brother, Abd al-Aziz. He was also a matrilineal great-grandson of the second caliph Umar ibn Al-Khattab.
Sadly, this name has NO meaning. Or, alternatively, the original meaning was lost. A look through the Arabic dictionary finds the simple definition that UMAR is a name, no meaning attached. There are many many words constructed by the same consonants ("umr" as you mention) yet there is no relation necessarily between these words, AAMER, AMAR, AMR, UMUR, AMMAR for example have completely different and unrelated meanings. So you cannot associate UMAR with any of these.