As part of the story I am writing, an individual is reanimated with relatively crude cryogenic technology. Given the constraints of biology, what would this entail in terms of real/current (or hypothetical/future) technology in order to be successful?
Well, we can't do it now so the only answer we have is 'more than we have now'.
Alright, this isn't strictly true, insofar as we can cryogenically freeze embryos and the like, and we can then thaw them and bring them to term so depending on your definition of 'someone', arguably the minimum technology someone needs is the ability to create and maintain liquid nitrogen. For full fledged human beings however, the problem is a little more complex.
When you get down to it, the problem is that around 70% of the human body is water. That may not sound like so much of a big deal - we freeze water all the time, right? Well true, but it's the stuff around the water that presents the problem here.
Ever wonder about the Titanic? Why it was possible for it to hit an iceberg in the first place? I mean, water is a liquid and most liquids are less dense than the solid made of the same material right? But icebergs float. So do those little cubes we put in our Scotch. That's because water expands when frozen, unlike most other molecules and compounds. Suffice it to say that freezing a complex human body actually causes massive damage to the cells because the frozen water actually damages the rest of the chemical compounds in the cell. (This is a simplification, but functionally correct.)
So; either we find a way to freeze water in a way that doesn't cause it to expand while the compounds around it increase their density, or we replace the water with another compound that doesn't react in that manner.
The first is simple - we simply don't have that technology and our understanding of chemistry leads us to believe that without some new breakthrough discovery in science, it will never be done. Therefore, it's not a simple engineering problem we can throw money and other resources at. It would rely on a scientific discovery to be made that we simply can't predict.
As for the second idea, most of the liquids we can think of that would behave in the right manner as they freeze are actually toxic to us, so while it's possible we wouldn't suffer cell damage, we'd still most likely die.
Bottom line is that in order to successfully cryogenically freeze a person with a high probability of bringing them back, we would either need to be able to replace all the water in all the cells of a human body with something that is both non-toxic AND subject to density increase during freezing; or, we need to find a way for water to react differently to freezing and increase (rather than decrease) in density.
Both are currently outside our known science, which is evident by one simple observation;
We're not doing it yet.
This is my favorite low tech method. Blood sugar and blood alcohol (and also acetone) are driven up in tandem. Way up. The person does not die of diabetic coma because her blood alcohol is so high, and does not die of alcohol poisoning because she is so cold that brain metabolism is slowed. You want the body no colder than the freezing point of alcohol. Before she goes to sleep, a left ventricular assist device is implanted; you can use the same line for a dialysis access.
When you revive her, you need to reverse all three in tandem: lower sugar, clear the alcohol and gently warm the body. Oxygen is going to be depleted too so you will need to bring that back artificially until the lungs come on line.
You could do that with current tech using ECMO dialysis. Combination of extracorporeal membrane oxygenation and continuous renal replacement therapy in critically ill patients: a systematic review.
Turn on the LVAD because the heart is too cold to do any work, and will probably not do much but quiver for a while. With the LVAD, blood starts flowing. Dialysis will clear alcohols in the blood and also glucose. You can gradually warm the blood in the machine too. With ECMO you can oxygenate blood on the return circuit but the high sugar content will support anaerobic metabolism for a little while. Lactic acid produced by anaerobic tissues will also be cleared by dialysis.
Once she is warmed up the heart will hopefully take over and you can turn off the LVAD. If not, gentle electricity will be salutary.
As Tim B pointed out it isn't technologically feasable yet to bring someone back from cryogenic sleep (sleep is a marketing euphemism since for all intents and purposes you are dead when frozen). As already said water needs to be removed due to it's freezing causing massive cell damage and other antifreeze fluids it could be replaced with will still cause a lot of damage. Additionally there is a limit on how long someone can be frozen due to the accretion of radiation damage in the body, which is caused by background radiation and the natural decay of radioisotopes within our bodies. In a living person the damage gets repaired naturally, but these repair mechanisms won't work in a (cryogenic sleeper) frozen, mummified corpse.
On that cheerful note let's discuss the technologies which would make (waking up) the reanimation of a long cold body plausible. Since cell damage is the issue at hand any technologie capable of fixing it quickly and on a large scale is fine. The obvious candidate would be medical nano-machines , be they biological or mechanical in nature. When we'll have these on a level capable of performing the complex tasks required to bring someone back from (a winter night dream) the land of the dead is yours to choose. Noone could argue against it if you place this level of sophistication somwhere between 2050 and 2200.
Consider that this level of medical technologie won't appear in a vacuum as beeing able to repair damaged cells at the rate needed for successful revival of a (sleeping beauty) slightly rotten corpse will make extreme longjevity or outright biological immortality possible. Going over Aubery de Grey's SENS approach   medical nanotechnology will fix all of the issues.
Finally this video by Youtuber Isaac Arthur goes into furtjer detail on cryogenics .
A good overview of the modern state of cryogenics is given in this article . There are frogs in nature which use glucose  as an antifreeze to replace dangerous water. While somewhat successful experiments haves been done with frozen rat organs none of the possesses is completly safe or reaches the temperature of 77 Kelvin where all biological processes would stop.
There are many possible developments in tech which would allow this, (OK , first, going along with the question's assertion that the freezing process has already occurred - thus excluding some modification akin to the adaptation in some frogs re. antifreeze in their cells):
Scanning tech close to the atomic-level - coupled with the ability to reproduce matter.
Basically, as soon as we develop teleportation tech, (and a high level of software editing to repair freeze-related damage) then we can focus those beams on the frozen individual, upload their physiological data (eg. all their brain's grey and white-matter data - all their life's experience, mental associations), plus their body (maybe with a few repairs/improvements). Then simply re-materialize them in their prime.
Time travel, coupled with mind-reading tech and cloning.
After death, their frozen remains can be discarded (they were only there for the family's viewing anyhow, PR style).
A person/bot is sent back to the most convenient time before death to retrieve the memories. For dementia patients, this could be pre-diagnosis, for murder suspects, just before the act (to ascertain mental state) or just after to get an exact view how it went. Memory editing by the state/third parties could be an issue in this society, I'd hate to be hacked - again. Anyway, the cloning wouuld produce a new body, an imprint of the mind would be uploaded to the blank-slate.
Nano, coupled with advanced AI.
Molecular level issues with cell integrity could perhaps be sorted out by the bots, reversal of blood clotting, cell membrane rupture, proteins being shredded by ice crystals. These minor issues could be repaired on an atom-by-atom basis (see "The World's Smallest Movie"). They could be repaired if the process is supervised by a splendidly intelligent AI, in communication with all the bots-in-the-bod, supervising the various stages of repair and reanimation.
One of the posts above talked about using alcohol. I'd propose we Cryo-freeze folks by first giving them genetic therapy that grants them traits of koi/carp. Carp, according to this article have the ability to survive in icy water by turning lactic acid (what happens when you exercise) into alcohol. https://www.foxnews.com/science/scientists-discover-how-some-fish-survive-in-icy-water
*> Per the BBC, goldfish and crucian carp developed the ability to
survive months in icy lakes and ponds using the unique ability to convert lactic acid into alcohol.
As water freezes and oxygen levels dwindle, lactic acids produced from eating carbohydrates are unable to escape a fish's body. These trapped lactic acids would kill the fish in minutes were it not for an evolutionary trait that allows them to convert the acids into alcohol, which is then released through their gills.*
This is where the scifi twist happens. This genetic therapy also gives our characters GILLS! This allows our characters to be frozen at near cryogenic temperatures while retaining the ability to breathe under water.
Folks wishing to prolong their life our stored under water in a cold facility or climate (free).
The drawback here and the low cost method of defrost would be that these folks would remain near "merfolk" until they could afford the genetic treatment to become an air breathing human again. Just like Amazon Glacial cloud storage, you can put your data (or body) in for free but when you want to reverse the operation it's going to cost you.
A method of the plot could also be how the cryo company raised rates and now have a ransom group of merfolk from 100 years ago.