Your sentence is grammatical. Pete's improvements are excellent. A weakness of your sentence is that "more than a little bit" is a cohesive thought that should remain intact. The word "luck" breaks those words apart, and that weakens the sentence. Adding "just" will improve the sentence either way: "I need more than just a little bit of luck." (best) "I need more luck than just a little bit." (satisfactory)

The sentence as written is ungrammatical. It should be written: "I need more than a little bit of luck." And yes, this means "I need plenty of luck/good fortune."