for the life of (one)
Despite one's best efforts; to any degree whatsoever. Often used in negative constructions. For the life of me, I can't remember where I left my keys! She coudn't understand for the life of her why people were so obsessed with celebrity gossip.
for the life of one
Although trying hard, as in I can't for the life of me remember his name. This expression is always used hyperbolically, that is, one's life is not at all endangered. [Late 1700s]