There may be many reasons for your dog to have this problem.
Why might your dog dislike rain so much? There may be several reasons. One of them may be a lack of exposure. Many dogs aren't used to being exposed to the rain when they are puppies so they grow up disliking the rain and the feeling of getting wet. Sometimes, the cause may be negative associations. If your dog is tentative in going out when it's raining and you scold him, your dog may start associating the rain with bad things. Dogs easily pick up our moods and reactions, so they may learn that rain is bad and should be avoided at all costs. For those who live in dry climates, when the rain comes in spring or fall your dog aren't too accustomed to seeing this much rain for a good part of the year. Dealing with this problem first hand here is a nice list of tips to help your dog become more comfortable in going potty in the rain.
Seven Tips to Help
1) Get your dog used to water
It is possible that domestication has spoiled our dogs. Dogs in the wild must get wet if they want to eat. Some domesticated dogs rarely get wet nowadays. I recommend getting young dogs used to getting wet from a young age. I add getting a bit wet in my socialization classes. Get a garden hose and let your dog play with the water. Your dog will have lots of fun!
2) Give food or treats on wet grass
Rain is often not the only problem with dogs who hate rain. Many dogs seem to also hate walking on the wet grass. You can classically condition your dog to love wet grass, here's how. If it has not recently rained, lightly make the grass wet with a garden hose and put your dog's food bowl down in the wet grass or toss some treats around. You can play a fun game of fetch on the wet grass. Your dog will start to associate wet grass with food or other fun things, or at least will learn that wet grass is nothing to worry about.
3) Take them for a walk
Walking stimulates elimination. Take your dog for a walk in the rain. Sooner or later the urge will come, so be patient. It may be tough at first but they will get used to the rain and gradually getting used to it will shorten the time to go potty.
4) Get a large umbrella
If you are looking for short cuts this may work. Doggy boots may be helpful as well as would a dog raincoat if you don't have a very large umbrella to cover your dog then get out your raincoat and rain boots. Get your dog a nice raincoat too!
5) Find a covered spot
It's not a bad idea to take your dog to a covered area to potty. Look for a tree with thick foliage, a carport under dirt or build an area that is covered but that has still dirt under that you can bring your dog to when it rains.
6) Wait until your dog really needs to go
This may work in some cases. If it is raining outside, delay your dog's potty time. If you have a regular schedule for your dog to go potty, then wait a while longer when your dog goes to the door. At this point, the urge to potty may override the fear of the rain.
7) Teach your dog a "Go Potty" Cue
This will help your dog generalize the behavior of going potty to different situations, places, and scenarios. This is a very helpful cue you want your dog to learn as it can make going potty a breeze.