This is set during the Renaissance, a mostly normal European setting.
Magic works through controlling the flux of energy. A mage can't simply create energy, but he can store and change it.
Most mages will meditate in front of water barrels, draining heat from the water until it turns into ice. A regular mage can generate about two barrels of ice per week.
Would this method be able to generate significant income for the mages?
There really wasn't an Industry for ice until 1805 when the first guy named Frederic Tudor tried to sell ice from new England in the Caribbean. Literally nobody wanted to buy it, and he spent 13 years trying to sell it and ended up in debtor's prison on 3 separate occasions trying to convince people that they even needed it. He was so obsessed with the idea of selling ice that he began bringing chilled beverages to entertain guests, who often scoffed at the idea of adding ice to a drink... until they tried it. He spent 20 years of aggressively marketing ice to people, convincing (even paying) bartenders to popularize chilled drinks, teaching vendors how to make and sell ice-cream, and convincing doctors it was required to cool down feverish patients. His idea FINALLY caught on around the mid 1820's and he ended up rich after two decades of introducing ice as a new cultural phenomenon.
The guy basically forced a meme in the age prior to internet. As odd as it sounds today up until that point people just really didn't care about drinking or eating cold stuff, there was somewhat a cottage industry in some cultures but it really wasn't something people wanted to pay for (packing root cellars with snow and such). Thing is, Ice as a product really didn't catch on in a big way until the 1840's, peaked in the 1860's, and was replaced by refrigeration in the 1900's. This all may seem like a distraction, but my point is people needed to be convinced they even wanted to pay for ice!
The average value of ice in the mid to late 1800's was 10 cents a pound, or about $2.70 cents today, ice vendors made their profits by selling anywhere from 10 to 50 tons of ice a season (with a total income per shipment of 54 thousand to 260 thousand dollars.) So a few barrels of ice a week wasn't worth much in the 1800's, (a few dollars) and wasn't really worth anything prior to the 1830's (nobody wanted it.)
But see, that's the REAL WORLD history of ice selling. You are writing a fictional world. So you got cash strapped mages who make ice as a byproduct of their work. So these mages decide to make up some snake oil, they start a rumor that "mage-ice" is super duper special compared to "normal ice." Chock full of good humors and invigorating spirits and what-not which are "rumored" to extend ones life, improve ones looks, and boost ones intelligence! (never mind the person who started those rumors is the ones selling the magic ice!) It becomes "fashionable" for Nobles to show off their clout by outbidding each-other and engaging in petty little squabbles to have a house mage on staff so they can have as much access to this wonderfully magical substance as possible. (in other words your mages do just what Frederic Tudor did in the 1820's and convince people its something they even really want to buy)
There really wasn't an Industry for ice until 1805. WRONG. Conservation of food has always been a use for ice, and certainly people did enjoy ice-cream and cold drinks. en.wikipedia.org/wiki/Ice_house_(building). A cuneiform tablet from c. **1780 BC** records the construction of an icehouse in the northern Mesopotamian town [...]. Certainly it was more limited due to technical issues, though.
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Would this method be able to generate signficant income for the mages?
Yes, it would. In a time when electric power was unknown, this would be the only viable way to have ice made easily available for many latitudes. The ice trade would start a few centuries earlier.
Ice created by magic would help preserve food anywhere in the world. Just this alone would be a major revolution. It would change the economy for food markets as well as for nautical exploration. Seriously, go read First Voyage Around the World by Antonio Pigafetta. After months onboard sailors would trade a king's ransom for a dead rat because it was the freshest edible thing available.
Ice has other uses as well. Medically, it can be used to treat rash and first degree burns, for example. Doctors would also use it to fight fevers. Barbers could press ice on a client's cheek after removing their teeth (barbers were the dentists of the past).
Gastronomically, asides preserving food, it is used in the making of ice cream. Ice cream is as old as the ancient romans, but you had to climb a mountain or wait for a snowy winter before the ice trade became a thing in the 19th century. If you think this is silly, remember that during the crusades people would kill for a pint of sugar.
People would find other creative uses for ice. If you want to take water upstairs, for example, you have to move the weight of the water + its container. With ice blocks, you don't necessarily need a container. And if you need to move heavy furniture around, perhaps placing it over a slab of ice makes things easier.
So long story short, I think not only the mages could make a living, they would be wealthy, upper class citizens.
TL;DR
The magician can earn a living quite well by producing ice, but don't expect him to get very rich.
We don't need to speculate. Natural ice trade is widespread and very well documented in medieval and modern ages -at least in Southern Europe-, and we just need to check for ice price to see what is worth the ice the magician can produce.
According to this paper (in Catalan) the official price to sell ice to the public in Estanyó (near Girona in Catalonia) was 7.5 denier/kg (cheaper than wheat, bread or wine), and that was about 1/6 of the daily wage of an unskilled worker.
According to the OP, the magician can turn two barrels of water into ice per week, although we don't know how big are those barrels. Assuming they are 100 litres barrels, the magician is producing about 1500 deniers/week, or the equivalent to 32 unskilled worker daily wages per week. That is, the ice he produces in a day is worth the wages of 5 unskilled workers.
From this amount we should detract the costs of distribution and taxes, since somebody needs to cut and sell the ice and ice was a heavily taxed industry.
I just used data from one place, which was just a village not far from ice sources. Probably in large cities more distant from ice sources the magician could get a better rate.
Anywhere there is a fishing industry will have a market for ice. So, large rivers, coastlines and islands. Ice gets its value there from preserving fish, allowing transport for a few days to markets inland.
In contrast to luxuries like cold drinks and ice cream, this puts it squarely into the realm of the essential - both for the inland city dweller whose life depends on adding protein to an otherwise poor diet, but also for the fisher people whose livelihood depends on getting their catch to market.
There's a distinct type of architecture known as an ice house, often built near the harbour pier in Scottish seaside villages.
You can push the timeline back a bit earlier than 1805 ... 1785 in Scotland, from a man who brought the idea of preserving ice home from travelling in China - you might follow it back to China to push the timeline back further still.
From the linked article:
At least 7,500 cartloads, or 2,000 tons of ice were needed to fill Berwick's ice-houses. During the 19th century, the ice needed was usually sourced from the River Tweed and harvested from ponds created in the district specially for the purpose, such as those at Heatherytops in Scremerston, about three miles south of Berwick. If sufficient amounts of ice could not be found locally, the Berwick Salmon Fisheries Company imported supplies from Norway. As early as 1833, there are records showing that ice was imported from Norway following a mild winter. The last ice-ship came to Berwick from Norway in 1939.
Having a local maker would be much more convenient than an international transport and storage system.
Ice houses quickly spread round the country from that first one, many still survive today though they fell into disuse when refrigeration came along.
The fact that custom buildings were worth building, insulated well enough to preserve winter ice through the summer, gives some idea of the real value of ice.
Ice was extremely popular in Renaissance Italy. It was added to wine to cool it down, used to keep wine chilled, both in the wine cellar and in the cup. Eventually a cup designed to hold ice or snow while another cup of wine was placed on top. They also used ice and snow to keep fruit, juice and other refreshments cold.
Ice cream would be another source of income. In Florence ice cream and sorbets were refined and extremely popular amongst those who could afford it. It was also used for special events among the rich in France and several other Southern and Central European countries.
So yes, money could be made as long as the mage is in a warm place.
Heck yeah! The ice trade didn't even begin until 1806. Before then, some ice (never large amounts) was captured from the Alps, etc., during the winter and stored (likely poorly) for summer use. Your mages would make a boat load during the summers (and not a penny during the winter), unless there are so many mages that it gluts the market.
Out of curiosity, though, why water? If the mages are drawing thermal energy from water, why not from fire, which is obviously more thermally dense? If you started with boiling water, the amount of energy drawn from two barrels to freeze them wouldn't be equal to draining a fire in a couple of minutes.
Would this method be able to generate significant income for the mages? Yes if he uses his brain.
Most of the population during Renaissance didn't have to worry about storing food, as they barely had enough to consume each day.
Things change if your mage operates in the surroundings of a court. And I suppose he already does, following Merlin example at the court of King Arthur.
In the past snow was collected and stored in cellars, and its use was mostly dedicated to prepare sorbets and ice creams for the courts. Your mage can enter this business, actually getting appointed as "official sorbeteer of his highness the of ".
There he can also use ice to improve storage of food needed for the court, and even provide ice for the times when somebody gets ill or hurt.
The above also suggests that your mage can have better fortune (read less competition) in southern countries where harvesting natural snow is somehow more difficult.
In this time period, ice is likely going to be popular with the very rich for its rarity and its ability to preserve delicate foods so they can be eaten out of season, especially if wizards are rare enough that someone's only going to be able to buy a few dozen ice barrels at a time and if there aren't any nearby tall mountains for easily-accessible snow.
Back in Roman times, the super-rich would hire fast chariots to speed the asparagus harvest (early Spring) from the Tiber river high into the Alps so that the snows would preserve the asparagus until the feast of Epicurious in Fall, and they'd pay insane amounts for the service.
The availability of ice in the middle ages would have changed society out of recognition. Having refrigeration would mean that every foodstuff could be stored and eaten at any time of year. It would mean that humanity would not have depended on seasons. Almost all the great festivals we have are to do with seasons. True many of them stem from a long time before the renaissance, but they are gradually being ignored and forgotten. Perhaps we would not even notice the coming of spring if we had been living with fridges for hundreds of years.
