"To show that homeopathy works would require evidence of approximately the same quality and quantity as the evidence* that concludes that it cannot work." (the "evidence*" in reference is that underlying the basic physics and chemistry that describes liquids, solutions, etc..)
This point cannot be emphasized enough. Homeopathy can't work unless basic findings in science are wrong. We have mountains of evidence for those findings while the best evidence for homeopathy is a handful of studies showing small effects. Anyone who says that a handful of studies showing small effect sizes is sufficient to move mountains is selling something.
The situation is much like a miracle claim, where the evidence necessary to prove a miracle must be such that "the falsehood of the testimony would be more miraculous than the event," to quote the Scot. The evidence necessary for homeopathy must be stronger than the evidence for basic science. I will explicate this idea in some detail. I hope my little rant here will illuminate the way science works in general and provide a bit of innoculation against pseudoscience.
So what are the mountains I'm talking about, and why do we have to move them for homeopathy to be true? (If you don't know what homeopathy is, I refer you to wikipedia or SBM http://www.sciencebasedmedicine.org/reference/homeopathy/
.) A homeopathic remedy is prepared by adding some solute to water then diluting the solution so much that the likelihood of a single particle of the initial solute being left is astronomically low. Since there's no longer any solute in the treatment, homeopathic remedies are literally just water. If you just want to know why homeopathy doesn't work, you can stop reading here. It's just water.
The homeopath has to quibble and say it's not "just" water. They will argue that water has a memory: the former presence of the solute has left some persistent mark on the water. But most data storage media (magnetic disk, DNA, brains) store data by making durable physical arrangements. As far as we know, water cannot do this. It's a liquid, which means the arrangment of its constituent H2O molecules is random. "Randomness" can be defined using statistical mechanics, and many deviations from randomness could be observed macroscopically. Even if the putative homeopathic non-randomness could not be detected experimentally, then how is it that stored information is carried to the human body and have health effects? Maybe we could detect it if we did the right experiment, Well, it's possible to invent ad-hoc rationalizations to answer this question. For example, the non-randomness could be physically in some way we have failed to detect. This could possibly be true, but we have no direct evidence to support it and considerable evidence to reject it. If you can think of another way that homeopathy could work that is meaningfully distinct from "magic," please let me know.
Note that if water has a memory, the homeopath has to explain why it remembers only what s/he added but forgets untold millenia passing through dirt, worms, bearded pigs and what not. Either that or explain why those memories don't matter. Since there's no more solute in the solution, how do you decide how much of it to take? Why do some solutions get diluted by a factor of ~10^-60 and others by ~10^-200?
What about the evidence claimed in support of homeopathy. There are many studies on the health effects, where you give sick people the remedy and see if they improve. (Here we can observe that if these people are sick, they should be given something that has some real chance of working.) The effects are generally modest, and "better" studies show smaller effects (better meaning more subjects, better controls, clearly defined outcomes, etc.). There is no direct evidence to support any mechanism of action. So okay, maybe there are health effects and we can study that. Considered in aggregate, most researchers looking at efficacy data conclude that homeopathy is a placebo. Some do not.
Keep in mind that the placebo-concluding scientists do not generally consider the violation of physics inherent in homeopathic claims. They just consider the evidence testing efficacy without considering the larger scientific context, which includes a much larger body of evidence that theories of homeopathy must explain. Even without that context, the case for homeopathy is very weak.
But lets consider all the evidence, not just the efficacy studies. On the one hand, we have mountains of evidence on basic physics and chemistry leading to unanimously accepted models suggesting that homeopathy cannot possibly work better than placebo, and we have the efficacy trial data that is ambiguous at best. On the other hand we have these few studies/analyses that show small effects, about which the medical community is divided.
So what theory better explains the evidence? On the "homeopathy works" side:
* the observed minor health benefits are explained by a working remedy
* the disconfirming meta-analyses of these observations are methodologically unsound
* inconsistent water memory is explained by gnomes
* the basic science is explained by some combination of rampant groupthink among physical scientists and special pleading/ad-hoc rationalizations
On "physics works" side:
* the huge body of observations of water and solutions is explained by existing physics and chemistry
* the small effects are explained by bias
It is very easy to show that various forms of bias produce "statistically significant" results where there is no effect under consideration (see the readable and free JP Ioannidis, "Why most published research findings are false," PLoS Med 2005).
The case against homeopathy is stronger than, for example, herbal supplements. There are many problems with supplements (notably, what's on the label may not be in the bottle), but, for example, pharmacologically active aspirin analogs are found in many plants. Moreover, the mechanism of aspirin's action was mysterious until the 70s, so we shouldn't be too aggressive in attacking treatments without a known mechanism. However, we know a lot about the body, and much, much more about isolated chemical interactions, so we can be pretty confident that, for example, homeopathy does not work.
Studies on homeopathy, reiki, etc. can only provide a veil of legitimacy to peddlers of snake oil. Stop doing them.