I have a container of dried out drywall mud and I wondered if there is anything I can do to make it usable again.
Drywall mud is so darn cheap, why would you do this?
That said, I actually tried this myself one time because I was temporarily without a car to get to the store, and I was impatient. I managed to get it to a somewhat workable consistency, but no matter what I did, I couldn't get the lumps out of it. I think if I'd let it sit and re-hydrate over a longer period, perhaps I could have worked out the lumps. But by that time, I'd have the car, and I just threw the lot of it away.
While you can rehydrate mud, the rehydrated stuff you make will not perform acceptably.
The reason is in the binder, the stuff that makes it harden and stick to the wall. This is usually polyvinyl alcohol (PVA). For reference, see the USG Sheetrock all-purpose joint compound MSDS, which lists "vinyl alcohol polymer" as an ingredient.
PVA is water soluble, but by the time it has dried, the polymerization reaction that makes it work as a glue has already happened. Getting it wet then somewhat reverses this reaction, but not completely. Even if you get all the lumps out, at a microscopic level you will still have a lot of polymerized PVA and other products of the reaction. You will essentially have drywall compound where a significant portion of the binder has already "dried", so when your rehydrated compound dries again, it won't bind as well as it should.
As a result, you will get poor adhesion to the wall, and likewise paint will not adhere well. This case study (now paywalled, see archive.org version) examines a case where the drywall contractors thinned excessively new drywall compound and got a similar result. While not exactly your case, it should demonstrate how the curing reaction in joint compound is not simple drying, and can be broken by deviation from the proper conditions.
Unfortunately you cannot re-wet drywall compound and get a usable product. It will break down in to a rough slurry, but the consistency will never be smooth enough to get a descent finish.
I have to admit, however, that's where my knowledge ended, so I had to ask myself... why? Why doesn't it just turn back into the mud from whence it came. So to satisfy my own curiosity I started googling.
Sadly, according to Google, to understand the "why" it seems you need a degree in chemistry, which I do not have. Basically there are chemical reactions in the setting type that cannot be undone by adding water, and there are additives in the drying type that gas off, chiefly ethylene vinyl acetate. Without replacing them you'll never get the same consistency. I suppose if you added those chemicals back in with the water you could get there but as Bobfandango pointed out, mud is so cheap why would you go to such lengths. I'm not sure what we learned here today but the short answer is: no.
I am a taper and you can rehydrate any of it and if it has chunks; you need to slowly add water to make sure the needed abrasion happens while getting mixed(if it becomes souppy; won't be quick). If it is 90,45 or 20; then very hot water after if possible having heated the the mud in the container and let sit covered in a hot area and the water will dissolve it and when you can mix it; add wood glue and use it as a taping mud.
I have rehydrated mud by adding water and mixing I am talking about premixed mud , hot mud can not be rehydrated, I used to buy mud in a dry powder form and ran out with only a small amount needed 30+ miles out of town I put in some water and used a drill powered mixer to rehydrate some old premixed added more water until it was the right consistency and it worked great, I do save the plastic sheet on buckets now and it never drys out or at least within a year or so. I use both paper tape and the nylon mesh depending on the area and have not had a problem rehydrating some but it takes a little time to get it mixed.
My understanding from a basic chemical point of view is: no, even if it looks similar it will be a different product without the strength or bonding qualities of the original.
Rehydrating set PVA mentioned in previous comments is problematic +1. Also the product contains limestone in the form of hydrated calcium (calcium hydroxide). When it is kept wet it stays in that form but in contact with oxygen it reacts and turns into calcium carbonate (also called limestone but chemically different). When drying into calcium carbonate it creates cure into the hard shape that it is molded into. Where as using calcium carbonate rehydrated solution there is no chemial curing, and the bond is weak, powdery and easily cracks, chips and crumbles.
It's the difference between a limestone block and a limestone powder mud pie.
To turn limestone (calcium carbonate) back into limestone (calcium hydroxide) you first need to add 825C of heat to turn into limestone (calcium oxide), and then add water through a process called slacking, which turns it back into limestone (calcium hydroxide).