Wait for a break or light rain - most dogs don't like to go outside during torrential downpours so wait for a break in the rain or at least when it is a light drizzle.
Misery Loves Company - the simplest solution is to put on my raincoat and boots, leash up my dogs and walk around the neighborhood. My dogs are willing to go out on a walk in the rain because, well, their walks are a positive experience and walking out in the neighborhood is always fun and exciting for them. I also believe that dogs are more willing to walk on wet sidewalks than wet grass. So, while it is not so fun for you to walk out in the rain, I find that this is the easiest way to coax your dog outside.
Put Peeing on Cue - the other thing I did when my dog Kiku was a puppy was to put peeing on cue by teaching her to pee when I said the words "go pee pee." On rainy days this helps keep your excursions outside short and efficient. Even if your dog is an adult, it is not too late to teach your dog this verbal cue. So this is how I taught my dog to pee on cue:
1) When you go on a walk, watch for signs that your dog is about to pee. For female dogs, it is when they begin to squat and for male dogs leg lifting is a good sign. When you see your dog begin these behaviors, say your cue (e.g. go pee pee, go potty, etc). Some people say "hurry up" or "go now" but I never liked that phrase because people have a hard time saying it in a positive tone. It tends to come out anxiety-ridden or impatient
2) When your dog finishes eliminating, say "yes" and reward my dog with a treat.
3) You will need to do this for a few weeks so your dog makes the association between the words, "go pee pee" and the actual act of urinating.
4) After a few weeks, test your dog to see if they understand what the verbal cue means. When you get close to a spot your dog likes to pee at, say the words, "go pee pee," wait 10-15 seconds and see if your dog begins to eliminate. If your dog eliminates, say "yes" and give a treat. If not, you may need to repeat steps 1-3 for a few more weeks before attempting step 4 again. Eventually, you will be able to tell your dog "go pee pee" and your dog will know that it is time to go.The above tips have been tried and true for me however, the following tips may also work for your dog:
Raincoats - some dogs do not like being rained on and a raincoat may help.
Wee wee pads - on very rare occasions where I had to leave the house for extended periods of time on a rainy day, I may leave a wee wee pad by the back door where they normally exit into the backyard.
Covered balconies and outdoor areas - for apartment and condo dwellers, sometimes it is helpful to teach your dog to go out on a balcony. Putting wee wee pads or boxes of sod can help encourage the dog to use this as an alternate elimination area but it helps if the dog is introduced to these areas early on when it is not raining.