The phrase implies hopefulness and positive thinking, especially after two unsuccessful attempts. He's tried that recipe twice and failed, but he's convinced it's going to be third time lucky.
Pro Tip 2/3
Casual Expression
Commonly used in casual or informal situations, rather than formal or serious discussions. I haven't caught a fish all day, but I'm feeling third time lucky on this next cast!
Pro Tip 3/3
Superstitious Origin
The phrase has superstitious origins and is often used humorously or lightly in conversation. Missed the bus again? Maybe you'll be third time lucky tomorrow!