To be very lucky. It is not a proper synonym, but in several cases it will bring similar results. :) (Sorry, I couldn't resist writing this)

'try my best' is more or less a synonym of 'do my best'. Obviously, 'try' and 'do' have different meanings, but in the context of this particular phrase, I think that they amount to the same thing. Neither of the other two phrases are quite right, I'm afraid. Two other possible phrases come to mind: 'do my utmost' - this is quite a formal phrase 'give 100%' - this is an informal expression