Be that as it may, the dialogue is quite sophisticated, logically speaking, and shows a higher-order awareness of how Plato's dialectical method, and argument in general, works. It is unsurprising that Aristotle, while critical of it, uses the dialogue as a way to discuss the logical features of induction.
The dialogue is structured much like Ion, is explicitly linked with Hippias Major, and has clear thematic links with Protagoras and Meno. The way of handling the themes, however, is unusual, and the nineteenth century scholars who questioned its authenticity did so entirely on grounds of its content. There is no definite consensus on how to interpret the work. It is also somewhat difficult to translate, since some of the key terms would have been very common for the Greeks but are tricky for us to handle without assuming things the Greeks might not have -- for instance, pseude, one of the most important terms and the one that gives the dialogue its common subtitle of 'On Lying', means 'liar' but in a very broad and loose sense (something more like in the colloquial English phrase 'make a liar of me', which implies that one did not strictly lie, but that events have intervened that made what you said turn out to be false).
You can read Hippias Minor online in English at Perseus Project, and in Victor Cousin's French translation at Wikisource.
(in order of appearance)
Eudicus is only known elsewhere from Hippias Major; he is an Athenian hosting Hippias.
Hippias of Elis
Hippias was, with Gorgias and Prodocus, one of the most famous teachers of oratory of the day. Besides the two dialogues named after him, he is also found in Protagoras and Xenophon's Memorabilia, and is mentioned in Phaedrus, the Apology, and Xenophon's Symposium. He was famous as a polymath, and this will play a role in the dialogue. By a very longstanding, and entirely possible, historical tradition, he is usually regarded as the discoverer of the quadratrix curve. At the beginning of the dialogue he has just finished giving a demonstration of his oratorical skills, taking Homer as his topic.
The Plot and The Thought
The dialogue breaks fairly easily into two parts, each distinguished by a deliberately shocking conclusion:
(1) The truthful and the false man or liar are the same man.
(2) The man who does wrong things voluntarily, if there is any such man, is better than the man who does them involuntarily.
I will call (1) the paradoxical conclusion and (2) the scandalous conclusion.
Eudicus opens the dialogue by asking why Socrates is so silent after Hippias's oratorical demonstration. Socrates responds that there are things about Hippias's discussion of Homer that he would like to hear Hippias discuss more, because it put him in mind of Eudicus's father's claim that the Iliad is a better poem than the Odyssey to the extent that Achilles is a better man than Odysseus. So Socrates asks Hippias which of the two men were better. Hippias agrees to answer questions because, after all, that's what he does. Socrates ironically remarks about how wise and confident Hippias is, to which Hippias replies that he has good reason to be confident, since he hasn't ever met anyone superior to him in anything.
Hippias's answer to the question is that Homer made Achilles best and bravest, Nestor the wisest, and Odysseus the wiliest. Socrates expresses bafflement, saying he can see why someone would say that Achilles was best and bravest or Nestor wisest, but what did he mean by saying that Odysseus was wiliest? Wasn't Achilles wily?
Hippias responds that Homer made Achilles not wily but the most guileless and truthful, quoting the Iliad (9:308ff):
Son of Laertes, sprung from Zeus, resourceful Odysseus,
I must speak the word bluntly,
How I will act and how I think it shall be accomplished,
For as hateful to me as the gates of Hades
Is he who hides one thing in his mind, and says another.
As for me, I will speak as it shall also be accomplished.
Thus, says Hippias, in these lines he lays out the nature of each: Achilles is truthful and guileless, and Odyssey is wiley and false-speaking.
Socrates moves the question for the moment from Homeric interpretation to Hippias's own claim, asking him if false-speakers have the power to do anything. Hippias says they do: the power to speak falsely. He asks is they are intelligent, and Hippias says they are: in false-speaking. This is the basic foundation.
Socrates then gets Hippias to agree that truthful people and false-speaking people are complete opposites, at which point he begins to argue against him, arguing that someone who is wisest in a field has the power both to tell the truth about it and to speak falsely about it. This is all Socrates needs to establish the paradoxical conclusion.
Hippias accuses Socrates of constantly quibbling and says that he will give Socrates plenty of evidence that Homer makes Achilles better than Odysseus, and that Achilles is not false-speaking and Odysseus is. Socrates goes on to give evidence from the very same passage Hippias quoted that Achilles said inconsistent, and therefore false, things. So Achilles and Odysseus are quite similar.
Hippias responds that Achilles when he speaks falsely does so involuntarily, whereas when Odysseus does it, he does so on purpose. Socrates argues that this is not so, but when Hippias continues to argue the point, he replies that Hippias now is arguing that Odysseus is better than Achilles. Hippias is baffled at how people who are voluntarily unjust could be better than those who are involuntarily unjust. Socrates says he is confused, and asks Hippias to cure his soul for this confusion; he appeals to Eudicus, and Eudicus remarks that he is sure Hippias will continue to answer questions. Hippias concedes this but only after saying that Socrates always sows confusion and argues unfairly.
Socrates then argues that in every other kind of case we take people who only do bad things on purpose to be better (more skilled and more able) than those who do bad things because they can't help it. But justice is a kind of power or knowledge or both, so someone who does injustice voluntarily is a better person than someone who does injustice because he can't help it. Or, as he puts it (376b): "it's up to the good man to do injustice voluntarily, and the bad man to do it involuntarily". It is good people who do injustice voluntarily. Thus the scandalous conclusion.
Hippias is unconvinced, and says he can't agree. Socrates replies that he doesn't agree with himself, but given the argument, it can't but look the way he said. And he ends the dialogue by remarking, ironically, that it is unsurprising if ordinary people like himself waver on the point; but it seems bad for everyone if they can't even stop wavering in the company of someone as wise as Hippias.
* We have from Porphyry a paraphrase of an argument from Socrates' student Antisthenes that discusses exactly the Homeric passage Hippias quotes, and at least raises some of the same issues. So some people have thought that perhaps this dialogue (and maybe Ion as well) was originally written by Plato as a polemic against Antisthenes; the two were quite opposed on a number of things. Unfortunately, we probably don't have enough of Antisthenes' argument to pin down exactly what would have been the point of contention. It's a salutary reminder, in any case, that Plato's dialogues had an original context, and when he takes a strange turn -- it might not have been strange at the time. It is also a reminder of how impoverished we are; only the works of Plato and Xenophon among the several major students of Socrates have survived more or less intact.
* It is probably not an accident on Plato's part that Hippias seems to get the interpretation of the Iliad passage he quotes incorrect -- while Achilles is talking to Odysseys, "he who hides one thing in his mind, and says another" is in context pretty clearly Agamemnon, not Odysseus. From claiming that he has never found anyone superior to himself, Hippias ends up being outmaneuvered by Socrates on Socrates' turf, outmaneuvered by him on his own turf, and forced to complain about how Socrates is unfair.
It is perhaps notable that Xenophon also depicts Hippias as complaining that Socrates does not argue fairly.
* Socrates' argument in this dialogue is often regarded as inconsistent with his argument in Crito; there he argues that we should never intentionally do injustice. But this supposed inconsistency is manifestly nonexistent; Socrates never at any point in this dialogue argues that we should do injustice. His argument here is not that we should do injustice, but that the only people who can do injustice in a way that comes from power and intelligence are just people. And it is worth noting that he says something very similar in the Crito itself (44d):
I only wish, Crito, the people could accomplish the greatest evils, that they might be able to accomplish also the greatest good things. Then all would be well. But now they can do neither of the two; for they are not able to make a man wise or foolish, but they do whatever occurs to them.
Likewise, people get very exercised about the scandalous conclusion, but, while paradoxically stated, there is a sense in which it is exactly right, exactly the same sense in which we can say that wrongdoing is a kind of weakness or foolishness -- recall the first few steps of Socrates' argument. But this doesn't necessarily make it any less paradoxical.
* People have noted that Socrates actually phrases the scandalous conclusion conditionally. In other dialogues (Protagoras and Gorgias) he gives arguments suggesting that injustice cannot actually be done voluntarily.
* Xenophon in his Memorabilia (IV.2.14ff, see especially IV.ii.20) has Socrates discussing much the same matter with Euthydemus -- so much so that either both Plato and Xenophon are drawing on a common source or one is adapting the other. However, there are a number of very importance differences. Most obviously, in Xenophon's version, Socrates is not perplexed about these things, only Euthydemus, and the whole discussion is preparatory to Socrates teaching Euthydemus plainly what things like justice are. That is, Socrates is bringing Euthydemus to recognize his own ignorance, and Euthydemus, of course, then concludes that he needs to learn these things from Socrates.
Quotations from Hippias Minor are from Nicholas D. Smith's translation in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 922-936.
Quotation from Crito is from Harold North Fowler's translation at the Perseus Project.