"May" is formal and / or polite. You can use it with your friends, but it's usually not necessary. You can also use it in business and formal situations. "Might" is extremely formal and / or polite. You can use it in business, although it's usually not necessary. Don't use it with your friends except as a joke -- they should laugh.

As a rule of thumb, "Might" is used for more hypothetical situations, or things that are remotely possible. May on the other hand, is used for situations that are more factual, or could be factual. So, the correct sentence would be "May we suggest going to a restaurant instead?" Since suggesting a restaurant is not a hypothetical situation. Okay now, we could make it into a hypothetical, as such: If I was to win the lottery, might we suggest going to a restaurant? Since winning the lottery, is a hypothetical situation, we used might. There are a lot more distinctions, with might and may, which I will shortly list: - Might is the past tense of May. Example: I may go to the movies today. I might have went to the movies, yesterday. The last distinction is when asking for permission, we use may more than might, but they can be used interchangeably: Example: May I go to the shop? Might I go to the shop? For an english speaker, reading those two, the version with may sounds natural, while the other does not.They are both correct.