Jacob is renamed Israel by God as explained in this passage:
Jacob was left alone. And a man wrestled with him until the break of dawn. When he saw that he had not prevailed against him, he wrenched Jacob’s hip at its socket, so that the socket of his hip was strained as he wrestled with him. Then he said, “Let me go, for dawn is breaking.” But he answered, “I will not let you go, unless you bless me.” Said the other, “What is your name?” He replied, “Jacob.” Said he, “Your name shall no longer be Jacob, but Israel, for you have striven with beings divine and human, and have prevailed.” Jacob asked, “Pray tell me your name.” But he said, “You must not ask my name!” And he took leave of him there. So Jacob named the place Peniel, meaning, “I have seen a divine being face to face, yet my life has been preserved.”—Genesis 32:25-31 (NJPS)
I've always heard that Israel means "God struggles", but according to Wikipedia:
Commentators differ on the original literal meaning. Some say the name comes from the verb śarar ("to rule, be strong, have authority over"), thereby making the name mean "God rules" or "God judges". Other possible meanings include "the prince of God" (from the King James Version) or "El fights/struggles". "The Jewish Study Bible" of Oxford University Press says on page 68 "The scientific etymology of Israel is uncertain, a good guess being '[The God] El rules.'"
All of these possible meanings seem to work with the story to varying degrees. Strong's suggests another possibility that would put an ironic twist on the story: "God prevails". Jacob prevailed until the man he was wrestling with cheated (though I suppose there are no explicit rules against using miracles as wrestling moves). Then the man gives Jacob a name that will remind him that God will always prevail, yet he is also told that he (Jacob) struggled with God and prevailed.
In the original, would this be the sort of story where the meaning is purposely vague? The name Peniel is spelled out in detail, but the more central name in the story, Israel, can currently be interpreted several ways with slight nuances imparted to the narrative. Was this by design?