If a player cast the spell Gaseous Form on a creature and, while the creature was in that form, it started raining, what would happen to the creature?
D&D 5e just isn't this simulationist
The question can't be answered by the published D&D rules, as the game is not meant to simulate such interactions in any way 'realistically'. The intent of Gaseous Form is to give various advantages of movement and protection to the user, and though there are limitations it is not meant to be accidentally deadly. Weather in D&D is often used to provide atmosphere, and what small space is given to mechanical details in the DMG does not cover this kind of hypothetical situation.
The spell description gives all the benefits and limitations expressed by the rules, and it says nothing about rain. Though the description says:
The target can pass through small holes, narrow openings, and even mere cracks , though it treats liquids as solid surfaces,
the intention of this is to allow the beneficiary to move like a gas, giving some benefits, but be stopped by solid bodies of liquids, like the surfaces of rivers, lakes, seas etc.
Rain in D&D
As a DM, I use weather mainly to provide atmosphere, and this approach is supported by some advice to DMs in the Starter Set:
Describe the party's overland travels as vividly as you like, but keep the story moving. "You walk for several miles and encounter nothing of interest" is far less evocative and memorable than, "A light rain dampens the rolling plains as you travel north. Around midday, you break for lunch under a lonely tree. There, the rogue finds a small rock that looks like a grinning face, but otherwise you see nothing out of the ordinary." (Lost Mine of Phandelver, 27)
Of course, the fact that the DM has described it is lightly raining can, at the DM's discretion have some mechanical implications (e.g. walls are slippery and so the DC is higher to climb them). But the only generally-applicable mechanical information about rain that I am aware of is in the DMG, p. 110:
HEAVY PRECIPITATION: Everything within an area of heavy rain or heavy snowfall is lightly obscured, and creatures in the area have disadvantage on Wisdom (Perception) checks that rely on sight. Heavy rain also extinguishes open flames and imposes disadvantage on Wisdom (Perception) checks that rely on hearing.
Note that this is only describing an extreme weather condition (heavy rain, not any rain), and in any case no mention is made of magical effects like gaseous form.
The Gaseous Form spell says that an affected creature treats liquids as a solid surface, but rain isn't a surface. Looking at the fluid dynamics of the question, I would imagine that raindrops would pass right through the target of the spell, and the gaseous creature would coalesce in their wake. From a role-playing perspective, it would probably tickle.
In 5e, there is no split between flavor text and rules text.
In the description of Gaseous Form, we are told that Liquids are treated as solid surfaces by someone in Gaseous Form.
This can be read a few ways. One interesting way to read it is that rain would be treated as a kind of hail stone -- the droplets would bounce off the creature (as if the liquid surface of each rain drop was a solid surface).
Another reading would be that the liquid drops wouldn't be slowed down by the gaseous form at all (as the form isn't very solid), but it would "tear" through the gaseous form. This could be interpreted as doing damage. I'd advise against it: the ability to flow through cracks/small holes and resistance to nonmagical damage implies that being "contiguous" isn't that important to the gas formed creature.
A mixture of the two also seems reasonable. Have the water go through the gas form creature, but treat it as if it was hail. So a torrential downpour might deal as much damage as a hail storm would to a normal creature (halved, due to resistance), which honestly isn't all that much damage. By the time the rain would get intensive enough to deal "real" damage, the wind would be more of a concern (both for gas form creatures, and non-gas form creatures).