We have grassy dog run on the side of our house- that is his usual potty area. If it has been raining, he will not go near the wet grass! Rather, he potties on the back deck- every single time!!! Totally drives us nuts!
The only thing is, she figured out that the sooner she goes, the less time she has to spend outside in the rain!
I pray for that day to arrive. 
I insist that mine pee even if it's raining but they'll stand for ages in the pouring rain hoping that I'll give way. No thought of peeing quickly and getting back in the dry, just stubbornly hoping that they can outwit me!
Lycia has a bladder of steel.. I have seen her go almost a full 24 hours with no peeing due to adverse weather. hehehe
When its just a light rain mine will crawl under the deck and go but if it's a downpour, then I have to pick her up and carry her down the block and make her walk back. She definitely wants to get home when I put her down, but the running to get home makes her want to pee. Mission accomplished. Be sure to treat big when she does it.
As they say, the key is do not come in till he goes, period. And you might want to consider walking him a bit… helps to encourage them to go potty
when we first got Kenji he didn't like the rain at all. He did the same thing. But I found that taking him out soon and offten in the rain got him comfortable in the rain… He still dosent voluntarily walk into mud or poddles or the rain but he realizes that it won't kill him.
first I started taking him out and if he doesn't pee he goes right back in his crate. After 15 to 30min I take him out and straight to the door. Ring bell and "hurry up potty" again. If he pees and or poops, praise, go right inside and he gets to play for a while. Also start walking him in the rain. When I walked kenji in the rain he would pull for a while and walk between my legs but keep walking and try to keep him to your side. Again this is what worked for me but whatever you do, do it all the time and consistently.
Lycia has a bladder of steel.. I have seen her go almost a full 24 hours with no peeing due to adverse weather. hehehe
LOL, Cody too, although if it had been up to him, he would have waited even longer!
When it's raining, he'll stand at the door for a few seconds like he's weighing his options. "Should I get wet and dirty, or should I just hold it?" When it's a heavy storm, he doesn't even pause, he just attempts to run back into the house. On those days I have to pick him up, carry him into the back yard to his spot, wait for him to do his business, and then carry him back inside. If I didn't do that, he would just stand there like he was paralyzed by the rain (his own personal kryptonite!). He has me trained well. Luckily we don't get many storms like that 
We live on the coast in BC, so Kipawa, who was born in the Pacific Northwest, has realized rain is a part of life here. Of course he prefers dry weather (what basenji wouldn't!), but if he really has to go, he will go in any weather. Certainly it does help if I am there, also getting wet, saying "hurry hurry". I believe that because he also sees me getting wet, he figures that the situation is then fair.
When it is raining, I have to take the dogs out 1-2 at a time and stand out in the rain with them until they go.
Well, it was storming outside, and I stood in the rain with him for about 7 minutes, but all he did was cry horrendously, run around in circles, pull on his leash, and try to hide under my legs to avoid the rain. 7 minutes in a downpour is a lot to me…lol. It just didn't seem like standing and waiting would have worked. I'm pretty sure I would have had to stand till he uncontrollably peed on himself lol, and who knows how long that would have taken. I still need to get an umbrella /sigh
Walking helps to stimulate the bladder. Usually if they have to go they will hurry to do their business if they know they will get to come back in. My boy will run out to the tree pee as fast as he can and then pull to get back in but that is because he learned if he doesn't we will walk until he does.
