What conditions would have to happen to re-animate a mummified person to come back to life with consciousness fully restored? What would the conditions of the mummification be and biological/technical challenges that would have to be overcome?
Since we've never even done that on non-mummies.
As Twelfth said, classical mummies have their brains pulled out. Freeze-dried mummies have their brains desiccated. Getting all of those back to a fresh, juicy state, just as they were prior to death is no small feat - even assuming nothing has been eaten by bacteria or other things (worms! maggots! yum!). But even that's not enough, still have electrical charge/whatever. We don't know 100% what makes the brain tick, nor how to access memories even from the fresh ones. Whatever that takes.
We will have to depart a little from current scientific thought to make de-mummification even possible. With the exception of many people of faith, many of us believe that consciousness is a product of the grey flesh behind and above our eyes. Experiments and observations suggest that damage to the brain has a greater effect on consciousness than any other non-fatal damage to the body.
But lets step back a moment, and consider that the people of faith are correct on this issue. Lets postulate that consciousness is a product of some non-corporeal spirit which connects itself to the body in some undetectable way, yet remains undamaged even after the death of that body. We might even postulate that the brain serves as a radio antennae for that spirit such that when the brain is damaged, the expression of consciousness through the body is diminished. The consciousness is not itself damaged along with the brain, but its ability to affect in and be detected by the real world is reduced.
If that is how consciousness really works, then reattaching a consciousness to a previously deceased body only requires that we build a new radio antennae that is tuned to the particular spirit which houses the desired consciousness. There is no need for that antennae to be made of grey matter. We aren't so good at building organic machines yet, so maybe we can find a metal or silicon circuit which can serve the same purpose. Similarly, we can replace all of the dead muscles with mechanical equivalents made of metals, plastics and rubber. We would probably want to keep the mummy wrapping to cover up our clockworks, but in the end, our spirit driven robot would compare pretty favorably to Boris Karloff's mummy.
With a very loose definition of "mummy" as "any person who has parts, or all, of themselves preserved in some way" and a future setting - cryogenics might be a plausible answer. Perhaps the technology was given up on, but they kept the sites running as a "tomb" of sorts.
Though, if it works and the cryogenically frozen can come back alive, I don't think most people would think the definition of "mummy" really applies (from today's viewpoint anyway.)
To expand more specifically to the OP's questions...
The brain needs to stay intact and preserved in some way. That's probably the most challenging part of this question to complete convincingly, its almost better off unexplained.
[These people] used some unknown chemical which preserves everything perfectly! The second half of this problem is removing this chemical or restoring the brain and body to where it didn't undergo whatever mummification process it went through, again without any sort of damage to the body.
To re-animate, there needs to be some way to start the brain and all the functions it controls. Electricity is probably the most used solution for this case, but it could be an injection of some type. The alternative is that reversing the mummification process resumes normal body functions automagically.
Does it have to be a human (Egyptian?) mummy? If it's some alien life form, or maybe even a genetically engineered human... well, these guys can do it: https://en.wikipedia.org/wiki/Tardigrade
You can obtain their DNA, thus you can rebuild their physical body (see "The Fifth Element" for an example).
So the only problem is whether you can recover the information they had prior to death.
Presumably you have access to the dehydrated and rotted remains of the brain. If you posit that the consciousness is held in the structure of the brain, and that a molecular scanner can scan the remains and detect the structures, then the only issue is whether the disassembled, rotted, dehydrated pile of brain molecules can be understood.
Shredded paper can be put back together like a puzzle. DNA is read by breaking it into pieces, reading the smaller chunks, then puzzling it back together.
So even if the brains were "scrambled", then a sufficiently powerful enough computer could figure out how it all fits back together.
So if you can feed it the molecular structure of the pile of rotted, dehydrated brains, and if it can see past the rotting or "repair" most of the rotting (we assume here that the structure is very redundant, and/or the rotting doesn't actually affect the way the neurons store memory) then in theory it can develop a model of the brain structure prior to death.
It then remains to rebuild or readjust the new brain so it matches the structure the mummified brain had. Then you start the heart, oxygenate the blood, and scream, "IIIIT'S ALIIIIIIIIVE!" while cackling gleefully.
The process of mummification would have to be significantly different from what it has been historically, as mentioned by the other posters and commenters.
Most importantly, it would probably have to start with a living person, rather than a days old corpse. We want the cells 'frozen' so to speak, waiting to be restarted, rather than literally dead as in a traditional corpse.
The purpose of the original mummies was eternal life. If you think about it this way, the mummification process resembles nothing better than cryogenics today. In cryogenics, a freshly dead body (or just head) is quickly cooled down to a very very low temperature. The goal of the operation is to prevent further decay. Unfortunately, current means of cooling bodies are destructive, in that ice crystals form that destroy cell membranes and make functionality at reheating essentially impossible.
So. How would a revivable mummy look like? I think the more important question is about its coffin. It would be a high-tech device (well, i suppose large granite slabs were high-tech 4000 years ago) that would maintain a constant (probably low) temperature. The body inside would have undergone the process while (barely) alive. Since we wish to restore consciousness, presumably we don't want to grab the brains out through the nose like the Egyptians did. If done very very carefully, we might be able to gradually thaw out the person days, months, years, or even centuries later. There are examples of (much simpler) organisms in nature that can revive after freezing, so the idea is not completely insane.
Even so, revival in the (same) flesh might be out of the question. More likely, future technology would be used to generate a digital upload, and even more advance handwaving would be needed to reincarnate that mind.
An emergency system, in a biotec/nanotech future, could be built which packs up the brain state when life becomes unsustainable due to environment or damage to vital organs (other than the brain itself). Suppose that it is destructive, so not good for routine backups. Nanobots or enginneered mechanisms in the cells first freeze the state, and then nanobots pick through it all, recording what was found as they take it apart. The results are consolidated into a compact and durable solid storage unit, for later physical recovery.
Once obtained and brought back to civilization, a body is grown and a brain printed out or somehow reset with the connections indicated by the store.