All videos of rocket launches I ever saw were performed in clear weather conditions. I also read many reports about launches which were delayed because of bad weather conditions.
What are the complications which would occur during a rocket launch when it rains? What modifications to a rocket could be made to make it possible to launch it anyway?
There is a huge difference between the US and Russian launchers, in terms of tolerance for weather. Additionally, different launchers have different tolerances.
The Russians launch from Baikanour, which is kind crazy weather wise, and cold, and snow, and other weather is much more common, so they designed from the get go for weather tolerance. It probably also helped that the R-7 ICBM became the Soyuz booster, and you don't get to delay nuclear war for bad weather.
The US side is much more delicate, in general, and much more careful about weather.
The Space Shuttle specifically had issues with launching and landing in rain. The tiles on the underside of the shuttle are very good at handling heat, but at the same time very delicate.
A rocket launching quickly goes supersonic, usually within the atmosphere, and usually within a minute or three of launch. Hitting individual rain, snow, or hail pellets at supersonic speeds is not a good thing. The tiles on the shuttle would have a real problem if that happened, alas as demonstrated by Columbia's final mission. Though that was a large strike, during launch.
In addition to rain, a bigger concern for long thin launchers is upper atmospheric winds. The amount of control a rocket has, depends on a number of things, but the vast majority of control is by gimballing the engine. If it is really windy as it ascends, it is possible that the engine cannot control the force of the wind, which could potentially push the vehicle into the air flow and destroy it. The Challenger orbiter itself was sort of destroyed more by the fact after the main tank ruptured, the orbiter tumbled in the airflow at multiple of Mach and was destroyed by the airflow than by the SRB having an issue or the external tank rupture (it did not explode, it burned) per se.
The issue however varies with altitude. The Shuttles in the end were suspectible to cold weather at launch ground level, as Challenger demonstrated. The Russians are much more cold tolerant since they knew what normal conditions were in Baikanour (vs Florida).
Once launched, your vehicle needs to be strong enough to handle supersonic impact of rain particles. Once high enough, you need enough gimbal control to handle winds.
How much of each you tolerate in your design depends on how much you care about robustness vs cost. It is a huge trade off. If you can build a much cheaper booster, but have to be more sensitive to weather, which way do you chose?
In the R-7 booster case, it was an ICBM that became a launcher, so it was more important to be able to launch on demand than cost, per se. (Of course, when you launch 1700 of them, the cost does tend to go down a bit). With the Shuttle, it was a pampered princess in this regard.
What are the complications which would occur during a rocket launch when it rains?
In addition to the issues raised by geoffc, another is that rain (or rather the rain clouds) obscure visibility of the rocket. Two key goals of launching from a US-based launch site are that the launch should not present a hazard to the US public at large, and perhaps even more importantly, that a US-based launch should not cause an international incident. A rocket veering out of control from Cape Canaveral has plenty of targets of opportunity for havoc: The communities around Cape Canaveral, Jacksonville, lots of cities on the coast. Internationally, a misbehaving rocket launched from Canaveral could theoretically hit Bermuda, the Bahamas, or Cuba.
The range safety rules for US launches are very strict with regard to weather conditions and the observability of the launching rocket. The rocket has to be visible throughout the launch, up to the point where it is high enough and moving fast enough that an onboard catastrophe would mean wreckage falling into the Atlantic.
Most of those old videos of rockets blowing up shortly after lift-off were a result of the Range Safety Officer pushing the "big red button" that sent an electronic self-destruct signal to a misbehaving rocket. (Actually, it was two small switches, both of which red to ensure they weren't triggered accidentally and both which had to be armed prior to moding to fire. Details, details.)
