They concern is actually for rider comfort - getting smacked in the face by rain going 70+ MPH is not pleasant. There is no official regulation or guidance on wet/dry roller coaster operation. Really, as far as I'm aware, is a temperature warning on not operating below freezing due to proper lubrication in the wheel bearings, etc.The rain rule is actually pretty smart. Why would you run a train on slippery track?
I have only ever heard of one documented case of a ride accident due to rain - Maverick's opening day at Cedar Point was an absolute down pour, and a rare moment when Cedar Point ran a number of roller coasters throughout the rain due to high volume attendance. Magnum slid through the first set of final brake run, bumping into the next train that was staged in the second set of brakes. Noone was injured gratefully, and the incident was moreso cited to an improper ride inspection by ride staff/mechanics than the rain.
Inverse to this however, I have been to Cedar Point a few times over the years during their Halloweekends event - the busiest time of the year for the park - when 100% rain was predicted throughout the evening. Rather than shut all the rides down, Cedar Point has, in these rare moments, opted to run the rides through the rain. And so, to my first point, I can confirm: Millennium Force is a bitch in the rain. Cedar Fair does remain pretty zealous with their rain rule however IMO, and can shut down operation far too quickly.
As Hixee pointed out, the biggest concern for adverse weather is lightning.
That is pretty correct - each park determines their own safe distance. For Cedar Point, it was roughly 20 miles, which was as far due west as Oak Harbor, Ohio - a fair guide for weather to expect. Since Cedar Point sits on a buffer between the Sandusky Bay and Lake Erie, weather patterns can be weird at times; rain storms sweeping along the coast, or storms staying in land.@Hyde will be able to confirm this for me, but Cedar Fair's policy takes this to the extreme. They will shut all rides if there's a storm detected within some distance of the park. Which makes sense really, as lightning can strike a few miles from the 'centre' of the storm, however Cedar Fair have set the distance at something quite huge like 20 miles? I think this is what Hyde is referring to.
A few years ago at Kings Island (2010 CF Live) I recall the rides getting shut down due to a looming storm. The storm was reasonably close, and sure enough there was a lightning strike on Diamondback during their test runs. Trains all then stuck on the lift hills and on the brake runs while they reset the whole thing. It was lucky there weren't any riders on board.
The more important rule, in order to avoid lightning strike hazard, is to stop all ride operations at the first sight of lightning, and not resume until 30 minutes had past. If a new strike of lightning occurs, the 30 minute wait restarts.
A random side note that I found to be true at Cedar Point when returning to operation from a storm: when Power Tower begins testing, every other ride in the park will begin testing within the hour/good chance of everything going fully operational.