A asteroid the size of Greenland, made entirely of loosely packed chunks of ice, in a decaying elliptical orbit.
Every few weeks or so, it brushes past the upper atmosphere and thousands of ~1m fragments break off. Those fragments burn up entirely in the atmosphere, increasing global humidity.
This would also accelerate global warming - water vapour is a greenhouse gas, as well as white clouds becoming dark rain-heavy clouds will decrease planetary albedo.
The extra water vapour will accumulate in the atmosphere, be distributed by high altitude winds, slowly descend, form clouds, and be eventually discharged as rain. Every few weeks the water is topped up by another pass of the asteroid, and whats left of the asteroid looses a little bit of speed.
After about 20-30 passes, the asteroid will entirely enter the atmosphere, but it'll be at a shallow angle, loosely packed, and almost entirely burn up before impact.
That should give the Earth a few solid years of rain.
Why do we need to bring the water in from space?
Using only water on Earth; This is really tricky, over water, you'll need to evaporate water while raining, water going back up and down again at the same time at the same place, carefully calibrated so that it doesn't push up too far (and build hail), or push up not enough (you'll just get a foggy mist, or it'll stop raining). You'd need to maintain this balance of wind and temperature overnight and through winter.
You're not going to be able to get it to rain in Antarctica easily either.
By bringing the water in from space in the form of evaporated ice, we can make it rain regardless of surface conditions.