A lot of dogs don't like to go potty in the rain. If that's the case with your puppy, you must train him to like it. Just take every opportunity to accompany your puppy outside when it's raining. If your puppy doesn't eliminate, put him back in his crate and try again 15 minutes later. Do this until he eliminates. Don't forget to praise him lavishly when he does. Another way to solve this problem is to go for a walk when it's raining. Since dogs like walks so much, your puppy will associate rain with an opportunity to be with you and he won't mind going outside afterward.
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Fundamentals of Dog and Puppy Training. Although dogs shouldn't be attributed with having human characteristics, they are intelligent enough to be able to understand the concept of, and execute, certain actions that their owners require of them - if these actions are asked in a way that dogs find rewarding.