I suppose this is a general question about the process historians use in deciding what order to study collections of ancient texts, especially when there are only a limited number of people able to translate them and too large a collection to get through in a reasonable time. With that in mind I don't want to necessarily limit the question to cuneiform but that's what I was thinking about when it came to mind.
For cuneiform tablets though, do they just look through them randomly until they see something interesting to them, like perusing through headlines almost? Or is there more of a system than that? I'm sure the location they were in found plays an important role in that but even within collections found at a single site there must be some sort of process. I assume most tablets have been digitized by this point, are people able to run keyword searches in those databases to find what they're looking for faster, even if nobody has actually read that text yet, or is that not feasible? Is it likely that there is important or groundbreaking information on a tablet buried somewhere in the collection that just hasn't been read yet?
I'll answer this question in different parts:
So we do indeed have lots and lots and lots and LOTS of untranslated cuneiform tablets. With that said, the number of people who can just pick up and (easily) read a cuneiform tablet are far fewer than you would think. For example, I am able to read transliterated cuneiform quite well (english renderings of the syllables/words/ideograms conveyed by individual signs). However, to read from a tablet is a far more difficult skill. Even among folks who read cuneiform quite well, they often read on previously drawn hand copies drawn generations prior. So in terms of assessing whether or not a particular tablet is important, it can be tough to assess just from looking at it.
Tablets are, by nature, often highly fragmentary. While you'll see some very large, impressive, mostly whole texts in museums, such as Sennacherib's annal known as the Taylor/Chicago Prism or large monumental inscriptions (ala those of Sargon II or Esarhaddon), most cuneiform tablets are fragmentary, broken, little hunks of crap. A few people in history, such as W.G. Lambert or Irving Finkel are/were known for their ability to find "joins"; that is, being able to place two or more fragments together to create a single text. Finkel has a story discussing this in his recent-ish book on Noah and the Flood. With that said, we often have broken little pieces of tablets that clearly have cuneiform on them, but without context or full lines, it's very hard to know what to do with them or even what they say.
Many tablets are indeed digitized, but nowhere close to all. Assyriology (and it's siblings, Hittitology and Ugaritology) are actually pretty solid in terms of digital humanities. They're one of the more advanced subfields of the humanities in this regard. Websites like ORACC (the Open Richly-Annotated Cuneiform Corpus), the CDLI (Cuneiform Digital Library Initiative), ePSD (electronic Pennsylvania Sumerian Dictionary), the ETCSL (Electronic Text Corpus of Sumerian Literature) and the free-to-download, totally searchable CAD (Chicago Assyrian Dictionary) make translating tablets and accessing texts easier than it ever has been. With that said, a lot of tablets are, to most people, "boring." The vast majority of cuneiform tablets are small economic texts, basically receipts. They're pretty useless even to someone like my self, who works primarily on mythic and theological texts from ancient Sumer and Babylon.
There is 100% earth-shatteringly important information on tablets that still remain in the ground. We have evidence for mythic stories that are, as of yet, undiscovered. For example, in multiple texts, there are references to a series of victories by the god Ninurta over various monsters. We have discovered on of these myths (bin šar dadmē, also known as the Anzû epic). There is a real possibility that similar full myths exist for each one of these enemies defeated by Ninurta. We just haven't found them.
Accessing the tablets themselves is notoriously difficult. Many tablets were acquired by museums through...spurious means to say the least. Various groups also hoard tablets to that only their lineage of students can access them. For example, good luck accessing most unpublished tablets from Mari if you're not French. They've had a stranglehold on tablets from Mari since their initial discovery.
Cuneiform studies is incredibly interesting, but reading cuneiform itself, as opposed to transliteration is tough, slow work.
Some suggested reading/viewing:
*The Oxford Handbook of Cuneiform Culture, edited by Karen Radner
*Browse through all of the information pages on Oracc
*Check out the Youtube page "Digital Hammurabi." They're good people and very well credentialed (husband and wife team, both of whom have degrees in NELC from Johns Hopkins)
Maybe a quick follow-up to u/rcxheth's really excellent answer:
A large number of the known cuneiform texts have at least been looked at by an expert at some point (museum staff, excavators), so often they are catalogued at least in general terms and at least the museums with smaller collections are aware of roughly what kinds of texts they have in their collections.
A trained Assyriologist can, generally speaking, find out some basic information about most texts relatively quickly from examining the text more closely. That is, it tends to be relatively easy to figure out what language a text is written in, what genre the text belongs to (an economic document can be recognised by the presence of numbers, a letter by specific greeting phrases etc.) and what time period it roughly dates to (usually by looking at the sign forms and orthography, but if you're lucky, there's a date in the text). This is of course significantly more difficult to do if the text is fragmentary or the surface heavily eroded.
There are different things one can do to 'publish' a cuneiform tablet. From just cataloguing the text (what genre is it? when does it date?) to making it available in a photograph and/or hand copy and/or transliteration to creating a proper text edition (making a hand copy of the text, adding a transliteration, translation and discussion of the text – definitely the ultimate goal, but also far more time-consuming than the other options).
Regarding how people choose texts for publication: very often, primary text publications are centred around provenance (texts that were excavated at the same place, sometimes these are grouped, for example, according to excavation seasons or find spots), museum collections (all or some of the texts in a given museum, grouped, for example, according to time period or genre) or themes/genre (e.g., all medical texts dealing with liver diseases). That being said, there was (and still is) a lot of cherry-picking going on, with scholars publishing the most interesting texts first. Plus, the vast majority of texts are administrative documents that are – as u/rcxheth also pointed out – quite boring if you look at them individually (and I say that as somebody who specialises in economic texts).
Next to the literature that was already suggested in the other posts, I'd also suggest having a look at the Cuneiform Digital Library – it contains entries for more than 300.000 cuneiform texts and information on what we know about them and where and when they were published, including high-quality images. The search function is not very intuitive, but you can search, for example, for genres ('letter') or places ('Babylon'). This should give you an idea that we actually know quite a lot, even though not all the texts are published.
And if you want to go very hardcore, then browse the Database of Neo-Sumerian Texts, a database of more than 100'000 economic texts dating to the time period between 2100-2000 BCE. These are just published in transliteration (not translated), but with wonderfully rich metadata and some very nice map and visualisation features. It will be hard to figure out what it's about but it will give you a good idea of what can be done and what many scholars are currently working on to making the texts more accessible to everyone.
A little disclaimer, I'm not an archaeologist, nor do I specialise in epigraphy. I also don't work in a museum or with cuneiform archives, so I am not the best person to answer all these questions. But I'm giving it my very best shot!
Depending on who you ask, digitizing cuneiform texts to the extent you described would be helpful to one branch of scholars (those who prioritise the contents of the texts), but potentially a hindrance for the epigraphers who focus on the tablets directly in order to study how the languages and writing developed and changed. We're talking over 3000 years of continuous history, with multiple languages and types of cuneiform. As with other fields, research of various kinds is being conducted upon these sources. They're all valid, but with totally different angles and forms of approach. I think much like parchment or papyri manuscripts, clay tablets can be digitally scanned, their contents can be copied, but there is an inherent value in the artefact itself, and they cannot be wholly dismissed or removed from one's research.
There is a sizeable backlog in terms of published translations, but each collection has been extensively catalogued. There are scholars working with these archives on a daily basis, going through them with fine-toothed combs in order to reconstruct broken tablets from collected pieces, as well as conduct their own research. They're not forgotten or locked away out of reach!
Provenance is a crucial element to our understanding of any text. The main priority from an archaeological and historical viewpoint is to know exactly where a tablet was found in a site. Was it a domestic house? A government building? A palace, or a temple? Where in the building was it found? A text's physical location tells scholars a lot about how it was valued by the people who made it/kept it. If we find an ancient archive or library, they are meticulously recorded and stored in the collections which their original owners placed them. In the first instance, scholars seek to understand the ancient people's priorities, rather than automatically impose their own upon the evidence. They are all meticulously dealt with equally, because one never knows when a receipt might contain vital information that helps you or another scholar to reconstruct something from the past.
As for the importance of some texts over others, I feel compelled to point out that even supposedly dry, "boring" ration lists or census documents are valuable mines of information, depending upon one's research.
Moreover, I have seen small fragments of cuneiform which have on them lines of the most exquisite poetry, which if only valued according to size and physical condition, would have instantly been dismissed. But if it were the only evidence from the entire site that someone had a copy of (for example) The Epic of Gilgamesh then it's a hugely important piece of data for the distribution not only of that story, but also for the kind of people who were living in that place. Therefore, each tablet is a valuable piece of data by which one can reconstruct and more deeply understand the past. By that I mean both because of what it says, and where it was found.
Provenance is critical. Modern archaeological methods are scrupulous about recording precisely where all manner of artefacts are found in relation to each other and any structures they might be in or near. This includes cuneiform tablets. If they were found with other texts, they are examined as a collection, and maintained as such afterwards. Or was it thrown away? Was it used as wall-filling in a later building? What does that tell us?
For the historian/linguist types, we often categorise them based upon academic conventions/traditions (i.e. 'types' of documents, such as 'literary', 'legal', 'administrative', etc.) These are due to the goals of certain types of academic research and do not reflect how the original owners of the documents viewed the texts. We take provenance into account whenever possible -- many weren't recorded in sufficient detail during early excavations. And we never dismiss how the ancients organised or prioritised tablets.
