Might typically falutes a little higher than may; since it's potentially counterfactual, it's more polite in a request, like using the subjunctive in Spanish: Dame un cafe 'Give me a coffee' is not rude (as it would be in English), but it's not nearly as polite as Que me de un cafe 'that you give[subjunctive] me a coffee'.
– John LawlerAug 20 '14 at 15:13
a very similar question was “May” & “Might”: What's the right context? http://english.stackexchange.com/questions/7/may-might-whats-the-right-context
– BrilligAug 20 '14 at 16:20
Since you seem to be specifically asking for permission you should use 'may'. If you say 'might' you're asking if the possibility exists but not for permission. (You visit your friend's house) "May I have a look around?" versus (you come home to your own home and the construction team is there doing some work) "Anything dangerous lying around? - might I have a look around?"
– BrilligAug 20 '14 at 17:50