What my son likes the most with winter is making a snowman.
What my son hates the most with winter is melting of his snowman.
I really would like to make him happy and make the snowman survive for as long as possible.
By the way, I am a king, with a medieval-level technology. My castle is in a temparate climate - think something like midle of France - quite far from the sea.
I already have caves and catacombs where it's quite cold, but I could build a specific building if needed.
There is no magic or alchemist involved, only snow. There is magic in my universe, but a realistic solution would be better.
What can I do to make my son's snowman survive, using medieval technology?
You could convert the water deposit of your main city in an ice cellar.
They were insulated with sand, straw, sawdust, but the water cisterns even made it colder. Imagine something like this full of water (this was the medieval cistern for a city with 30,000 people)
Just try this - even if I'm not sure if this answer fits your "no alchemy" point: Before building the snowman just mix some sawdust into the snow, the finer the better. This will effectively prolong the life of the snowman significantly without the need to build any extra buildings or the like.
See https://en.wikipedia.org/wiki/Pykrete for a sample.
P.S.: I would love to post any original sources here, but I can't seem to find any on the run which don't just coat the snowman in sawdust, not mixing the snow beforehand. But nevertheless this works just fine and lets your snowman stand in the open up until the sunny May or June.
In the region where I have grown up, they used to harvest snow during winter from nearby mountains and store it in underground rooms insulated with straw.
Build a large underground room, so that it is protected from the day heath.
Fill it with snow and straw, add salt for further cooling
let your son make the snowman, wearing a good woolen coat
give the snowman a woolen coat, too (yes, wool insulates from thermal exchanges in both directions...)
For the reference, Alps have snow line at around 2,5-3 km..
Put the snow man in a cooling bath with some salt in it.
Salts like sodium chloride lower the melting point of water-ice solutions: "−10 °C can be achieved with a 1 to 2.5 ratio by weight of calcium chloride hexahydrate to ice.".
That should keep the snow man cool. You would need a natural deposit of that salt. Mining will probably help there, because many million years ago there may have been sea where now is middle of France.
The knowledge of cooling by adding salt crystals to water would probably be something in the realm of an alchemist, but once somebody (a traveling alchemist) told you, the way to maintain a cooling bath would be rather simple (just keep on adding salt crystals and change the mixture often).
As a bare minimum, you should take a sprayer with water (or just broom/bossom; or even a bucket of water — depending on the snowman's size) and cover the snowman's surface. Of course, it should be relatively cold outside (so that water soon/immediately freezes). This may deform snowman's surface a bit, but probably will make the snowman somewhat more secure against temporary warmings.
Sorry, if the advice is too obvious.
Update: Per Nathan Cooper, it might be just a myth/mispractice/rumor that covering a snowman with a layer of ice would strengthen it against warmness. Thermal conductivity of ice is actually higher and albedo of ice is actually lower. Or this procedure (covering a snowman with a layer of ice) may serve some specific goals (like durability of small details against wind), but not the warmness-protection.
In addition to the cold-room solutions proposed by others, get your son to mix the snow with sawdust as he makes the snowman.
This will drastically increase the snowman's resistance to temperatures and melting by making a mixture known as Pykrete.
As an added benefit it's also much more resistant to physical damage, in case any pesky peasants attempt to destroy it.
Make a bell-shaped glass container, one for each snowman. Then with the aid of a little magic have a dragon suck all the air out of it in order to create a vacuum. A little more magic is needed to seal it. Then, similarly to interstellar space, thanks to the absence of particles that could transmit heat to the snowman, it should take quite long to melt. Keep it in the shadow, you never know.