I hope you might come home soon.
I hope you may come home soon.
What is the difference?
I would like a native speaker's opinion.
Do they mean the same thing?
I am not a teacher.
I don't feel that your sample sentences are very natural. I think you mean to say "I hope that you will come home soon."
Also you should know that "might" is the past of "may".
X says, "I may come home soon."
X said that he might come home soon.
When these words are used to express uncertainty or possibility they are largely interchangeable.
It is then a question of degree and in this respect "might" is considered by some to express less certainty than "may".
There are times when it could work- if, say, the person is in hospital and not free to leave, I could use I hope you may come home soon.
In the situation you just gave me, I could use either may or might without chancing the overall meaning?
I would use may, with the idea of being allowed to leave.