Is there a good example case where extant ancient literature has helped understand archaeological findings? This could mean, for example, a Roman author mentioning a tool and its use, which has helped us identify some archaeological findings that may have otherwise been left unexplained, or perhaps literature has helped locate or identify certain cities. The question in the title is broad, and I am not looking for an extensive description as whole. Instead, I am looking for a single good example from ancient Rome or Greece.
If you allow the answer to be a bit wider in scope than the question (this is not Physics.SE, and we have this leeway in humanities), and talk about how linguistic evidence in general, not only written or traditionally transmitted evidence, which is the scope of your question, I should mention a book that is considered somewhat controversial among archeologists, but strikes the right chords in me as a linguist.
The book in the spotlight here is: Anthony, David W. The horse, the wheel, and language : how Bronze-Age riders from the Eurasian steppes shaped the modern world. Princeton, N.J: Princeton University Press, 2007. The conventional archeology uses excavated evidence as the first class, and linguistic and other types of evidence are treated as circumstantial. Anthony's approach is exactly the reverse: He puts the linguistic evidence up front, and then beads the physical, unearthed data from the archeology on this string.
Do not get me wrong when I mention a “controversy.” Anthony is the first class archeologist, and there is nothing going on even close to the so called “fringe science;” this is just the approach to the evidence that is felt by some archeologists as a quite non-conventional. And to me, his picture of the ancient IE society looks quite compelling.
I am not trying to write this answer to review the book or retell it's story; it's hard to compress a monograph into a SE answer. I just want to mention his approach to the raw data, necessary simplified fo fit it in this format. Suppose you study the archeology of a culture that had words (according to sound linguistics) for “snow” or “bronze”--such that all, or nearly all daughter languages have inherited them . Already (be careful here, looking for parallel borrowings, though), you can start with a quite a strong hypothesis that the culture that spoke that language dwelt far off the equator enough to experience snowy winters, and their metallurgy knew bronze. This essentially excludes populations living in too lower latitudes or not showing archeological traces of bronze metallurgy from being associated with the speakers of the language in question. An example of two main concepts that are unquestionably made in in the daughter vocabularies are the ttile words for horse and wheel. Also, there is a great deal of other literary evidence of linguistic archeology is analyzed by Anthony, such as Rigvedic texts (and this connects it indeed to your question) that proves a very strong connection that archeology of physical things, namely burial practices. This also constitutes a very strong part in his arguments.
I would not personally say that the monograph is very accessible; far from being a pop-sci work, it requires you to dive into the references and learn a lot of things archeological (but as linguist, again, I am distanced far more away from that than the workings of language). But even without that, a claim to the very close match between literary Rigvedic burial prescription to those physically unearthed graves seems to me to be a very compelling evidence of the cultural connection
So even reading the work while skimming about the parts with a deep foray in archeology (which, I admit, could sometimes fly over my linguist's head) is indeed very enlightening.