As far as I can see, the actual physics of the car handling wouldn't need to be changed. The only change would be decreasing the friction of the track, no? The cars have mass and all the relevant forces, gravity won't change or anything just because it's wet. So that part could (in theory, without the graphics changing to show it) be done already with just some minor code changes.
If you ask me, the hardest part would be varying the friction across parts of the track, which obviously we don't currently have. Dry parts under bridges give more grip than wet parts, etc. Right now it's all 100% dry so no friction changes need to be calculated, nor does the track have to be split into parts (that doesn't mean it's not of course).
And because we want the wind to affect the rain, the wet/dry areas around these obstacles would move and change according to the wind. That's where it gets complicated.
I'm sure it'd be fairly easy to do a rudimentary "Friction -40%" calculation and throw a particle effect over the top of it to look like rain, but that'd be crap and go against everything the simulation stands for.