Rainforests have been named well, for they are associated with higher levels of rain than surrounding areas. The distribution of rainforests has been associated with geography (e.g. mountain ranges) and prevailing winds. But it's more than this.
In recent times, it has been discovered that these forests actually create their own rain. Many broad-leafed green plants expire water though transpiration, but they also have the capacity to emit aerobacter (bacteria) into the air.
Diagram One: Rainforest foliage emits aerobacter as well as water vapour through transpiration
These bacteria form nuclei for the formation of raindrops, effectively cloud seeding. Research has shown that, rainforest vegetation is an extremely efficient emitter releasing these particles high above the canopy.
Combined with the water vapour from transpiration and the creation of aerobacter, the rainforest creates its own cloud cover, sheltering it directly from prolonged periods of sunlight and even reflecting some of the sun's heat back into space.
This cloud seeding process, along with ocean and air currents function as part of a global air-conditioning system as well as being an important component of the earth’s water cycle. So indeed, rainforest is an appropriate name for these natural systems.
Diagram Two: Water cycle driven by rainforest vegetation