The dialogue does not give itself a clear dramatic date. However, it does mention Euthyphro the diviner. In fact, it mentions him four times. And among these mentions, at 396d, Hermogenes remarks that Socrates seems like a prophet (mantis) who has suddenly been inspired, and Socrates responds that he had been talking in the very early morning with Euthyphro, "lending an ear to his lengthy discussion" and been inspired by his superhuman wisdom. This occurs just after a discussion of Ouranos, Chronos, and Zeus, who were explicitly mentioned in the Euthyphro. Against this, many people argue that it's not possible for Socrates to have had the discussions in Theaetetus, Euthyphro, and Cratylus all in a single morning. Debra Nails has even more subtle prosopographical arguments against it.
However, I am convinced that it should be seen as a Last Days dialogue, and for a reason I have not seen anyone else give: the dialogue not only has the obvious apparent links to Euthyphro, it has obvious content links with Theaetetus, down to making some of the same points. In both dialogues, the issue of Protagoras & Heraclitus comes up as a crucial problem, and Protagoras and Heraclitus are named explicitly. And especially compare these two passages, first from Theaetetus:
This problem now, we have inherited it, have we not, from the ancients? They used poetical forms which concealed from the majority of men their real meaning, namely, that Ocean and Tethys, the origin of all things, are actually flowing streams, and nothing stands still. In more modern times, the problem is presented to us by men who, being more accomplished in these matters, plainly demonstrate their meaning so that even shoemakers may hear and assimilate their wisdom, and give up the silly idea that some things in this world stand still while others move, learn that all things ar ein motion, and recognize the greatness of their instructors. (180c-d)
then from Cratylus:
Most of our wise men nowadays get so dizzy going around and around in their search for the nature of the things that are, that the things themselves appear to them to be turning around and moving every which way. Well, I think that the people who gave things their names in very ancient times are exactly like these wise men. They don't blame this on their own internal condition, however, but on the nature of the things themselves, which they think are never stable or steadfast, but flowing and moving, full of every sort of motion and constant coming into being. (411b-c)
Earlier in Cratylus, Socrates had used Ocean and Tethys as one of his examples that Heraclitus was saying the same thing as Homer. When one adds to this clear confluence of ideas the references to Euthyphro's inspiration, the comments about the pious son of an impious father (394e) and the gods (400d-401a), various small similarities between Cratylus on the correctness of names and Euthyphro on piety, the direct introduction of Protagoras and Heraclitus, and various smaller things reminiscent of Theaetetus like the reference to geometry (436d; cp Theaetetus 180c) and the topic of knowledge (440a-b), then it seems that we are getting too many apparent links for none of them to be actual.
In any case, we will be reading it as a Last Days dialogue.
You can read Cratylus in English at the Perseus Project or in French at Wikisource.
(in order of appearance)
Hermogenes had a rich father, Hipponicus, and a rich brother, Callias, but was himself not wealthy, possibly because he was illegitimate, although Plato seems to suggest that he may have been unjustly deprived by Callias. He was Xenophon's source for Socrates' last days, and Plato makes him one of the students around Socrates in Phaedo.
According to a brief comment by Aristotle, Plato studied under Cratylus before he studied under Socrates; other than that, almost all that we know about Cratylus is in this dialogue. There is a story, however, that eventually Cratylus came to the conclusion that the only way to communicate anything was to point to it.
The Plot and The Thought
Hermogenes opens the dialogue by asking Cratylus whether they should have Socrates join their conversation, and Cratylus permits it. We then discover that there are two opposing positions on the table.
(1) Cratylus holds that there is an intrinsic correctness of names, independent of convention.
(2) Hermogenes, on the other hand, holds that names are purely a matter of convention so that there is no correctness of names.
Socrates will argue that there are insuperable problems with both of these. of course, there is a position that is neither of these: naming involves convention or custom but there is a correctness associated with it, and in fact this is what Socrates will end up arguing -- more precisely, he will argue that if names are for instruction, they must involve convention but also be subject to standards of correctness based on knowledge of things to which they apply.
In response to Hermogenes, Socrates asks if he accepts the Protagorean idea that man is the measure of all things. There is no non-arbitrary distinction between private and public conventions in a matter like naming, so Hermogenes's position seems to come to something like it. Hermogenes doesn't accept Protagoras's position, however, although he has difficulty not falling into it. He agrees with Socrates that we use names as instruments in order to "divide things according to their natures" (388b). On the basis of this Socrates argues that names require a nomothetes, which could be translated as 'lawmaker' or as 'custom-maker', and that the custom-maker must be supervised by the dialectician, since it is the dialectician who actually divides things according to their natures. They discuss a large number of names -- Socrates claims to have inspiration derived from Euthyphro -- and in the course of this Socrates remarks that the names seem to suggest the idea of Heraclitus and many modern thinkers that everything is constantly in flux. Socrates after this turns to a discussion with Cratylus -- who is, in fact, a Heraclitean.
Cratylus also agrees with Socrates that names are for instruction, and Socrates shows that this causes problems for the Heraclitean view, since it seems that if names are able to communicate there must be something not constantly flowing. Exactly how this works is not explored here, but Socrates gives us a good idea of what he has in mind:
But if there are always that which knows and that which is known, if there are such things as the beautiful, the good, and each one of the things that are, it doesn't appear to me that these things can be at all like flowings or motions, as we were saying just now they were. (440b)
Socrates remarks that Cratylus must investigate these matters very carefully. Cratylus promises he will, but says he has already given a lot of thought to these matters and is sure that Heraclitus is more or less right. Socrates seems to regard this as a reason to end the discussion, and tells Cratylus to tell him all about it when he gets back from his trip, and Cratylus ends the dialogue by saying that he will, but he hopes Socrates keeps thinking about these things, too.
* In the dramatic timeline, the next dialogues are Sophist and Statesman, in which a dialectician, the Eleatic Stranger, shows how to divide things in an attempt to get names to express natures correctly, the need for which is expressly argued in this dialogue. So if we take Cratylus to be a Last Days dialogue, it directly contributes to the sequence of thought that begins with Theaetetus. In addition, the dialogue can be seen as explaining the problem with the charge against Socrates, impiety: as we saw in Euthyphro, there is good reason to think that the Athenians -- as represented by Euthyphro and Meletus -- are not holding themselves to standards of correctness in names like 'piety' and 'impiety'. The reason, or at least a diagnosis, for this is given in this dialogue, as well, and links the dialogue to Theaetetus: the subjectivism of sophistry (as represented by Protagoras) and excessive focus on the flowing sensible rather than the stable intelligible (as represented by Heraclitus) is interfering with the knowledge required for correctness in naming (and thus in teaching, for which naming is an instrument).
* This dialogue has several puns on Hermogenes's name. 'Hermogenes' literally means 'son of Hermes'. Hermes is the god of profits, so the fact that Hermogenes is unsuccessful in monetary ventures is a reason why Cratylus might say he is poorly named. Hermes is the god of matters dealing with speech, and so this is also perhaps a reason why he is poorly named, since he is not good with speeches. But Hermes is the Psychopomp, Hermes Pompaios, and Socrates' last statement to Cratylus is that Hermogenes will go with (propempsei) Cratylus, a related term, so the dialogue ends happily with a joke that Hermogenes is appropriately named after all.
* Philosophers over the past century have tended to regard Cratylus as a baffling and mostly useless dialogue, which I think establishes conclusively that philosophers over the past century have been lacking in self-awareness. Of all Plato's dialogues, this one is the one that most directly touches on the kinds of philosophical issues that became central to philosophy in the twentieth century. On the 'analytic' side, the 'linguistic turn' was precisely motivated by the general kind of idea for which Socrates argues here: when it comes to conveying ideas, there is a standard of correctness by which one corrects error and misunderstanding. What is more, analytic philosophers have always struggled with the problems that arise in both engaging in this kind of dialectical activity of correcting names and also being inclined or tempted to Hermogenes's position that language is purely conventional. On the 'phenomenological' side, Heidegger can be seen as accepting the general kind of idea for which Socrates argues here, as well; but this side of twentieth century had the difficulty of being already inclined and thus at times tempted to a modern form of Cratylus's position, namely, that the understanding conveyed in language is rooted in temporal flow. This characterization, of course, is a crude representation; but one will find that it nonetheless goes quite far even without refinement.
* The perceptive reader will, I think, notice that this dialogue is in essence arguing for the viability of real classification, since that is effectively how all three participants in the dialogue are understanding naming. If you want a good way to see why the argument here is not merely some weird discussion of etymologies, think about scientific taxonomy, such as was done by Lavoisier or Linnaeus and was discussed by William Whewell. Whewell's eighth aphorism on scientific terminology conveys exactly the sort of correctness of names at issue in this dialogue:
Terms must be constructed and appropriated so as to be fitted to enunciate clearly and simply true propositions.
Scientific classification is a form of naming. It is not merely conventional, because it is held to standards of correctness; it does not naively follow the assumptions of ordinary language but is dialectically examined and corrected. Scientific classification, in other words, presupposes exactly the points made by Socrates in this dialogue.
* The main stumblingblock most modern readers have in their reading of the dialogue, of course, is the very, very long discussion of etymologies. But it's important to note that there is more going on in this discussion than just fanciful (and, note, Socrates himself flags its fancifulness at several points) playing around with syllables. In the course of doing thi, Socrates also incidentally does several other things. (1) He shows, rather than simply argues, that the correctness of names cannot be a matter of mere syllables but must somehow be a matter of understanding the natures of things themselves. (2) He prepares the discussion for the worries about Heraclitus that will come about later, by repeatedly introducing the assumption that all things flow, and in so doing also anticipates the discussion he will have with Cratylus. (3) He shows, regardless of the correctness of the etymologies, that both Hermogenes and Cratylus are wrong, since exploration of names requires attention to those things to which names are applied (pace Hermogenes) and yet also implies the need for critical care (by what amounts to a reductio by anticipation of Cratylus).
* Since we are reading this as a Last Days dialogue, the following passage in the discussion of Hades seems particularly relevant:
The words Hades knows how to speak are so beautiful, it seems, that everyone--even the Sirens--has been overcome by his enchantments. On this account, therefore, this god is a perfect sophist, and a great benefactor to those who are with him. So great is the wealth that surrounds him there below, he got the name 'Pluto'. On the other hand, because he is unwilling to associate with human beings while they have their bodies, but converses with them only when their souls are purified of all the desires and evils of the body, doesn't he seem to you to be a philosopher? For hasn't he well understood that when people are free of their bodies he can bind them with the desire for virtue, but that while they feel the agitation and madness of the body not even the famous shackles of his father Cronus could keep them with him? (403d-404a)
Quotation from Theaetetus is from Myles Burnyeat's revision of M. J. Levett's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 157-234.
Quotations from Cratylus are from C. D. C. Reeve's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 101-156.