Just now a fellow user used this "For the life of me" in an answer to my question. I'm not aware of this phrase but it seems incorrect to say "life of me". Shouldn't it be "life of mine"?
Yes, SovereignSun, it should technically be "for the life of mine", but it is an idiomatic expression, used colloquially often, for a dramatic effect. I often say "for the life of me" even though it is technically incorrect grammatically. No one would ever say "for the life of mine" in the context you've just heard it used in, however. It's just an idiomatic expression, so the grammar doesn't have to be correct. I would use "for the life of mine" if I were talking about my actual life, rather than being dramatic idiomatically:
"For this life of mine is but one life of the many that I have lived before and shall live after." (I'm making an argument for reincarnation here.) (I also use "this" instead of "the" as it's hard to use "the" when saying "the life of mine"; it's also much easier just to say "my life".)
I hope that might have helped you out. Just remember that idioms don't have to make sense grammatically; that's the reason they're called "idioms", which derives from a Greek word meaning "to make one's own".