Difference between ''may'' and ''may well''

Solution 1:

It may rain.

This means that rain is a possibility; it may happen or it may not.

It may well rain.

This means that rain is a possibility, and that the speaker thinks that there is good reason for it to occur.

For example:

You'd better bring an umbrella; it may well rain today.

Dictionary references:

Merriam-Webster Online Dictionary:

may well
10b. in all likelihood: indeed it may well be true

Cambridge Dictionaries Online:

may well
If you say that something may well happen, you mean that it is likely to happen:
She may well not want to travel alone.