What is the origin of the phrase "to give up"
"Give up" is almost a word-for-word translation of surrender (sur = over/above/up + render = to give, present), so it seems to be an Anglosaxon version of a French/Latin word. It has been used for much longer 400 years but was probably popularised through its use in the King James Bible:
30 When Jesus therefore had received the vinegar, he said, It is finished: and he bowed his head, and gave up the ghost.
John 19:30 King James Version (KJV)