First things first. Is there any evidence that homeopathy is effective for treatment of eczema? A quick search on Pubmed showed this to be the most recent (2003) relevant review. Incidentally, Pubmed is a freely accessible service, unlike many of the journal databases used by academics, and anyone with an internet connection can do the same thing as me and come up with the presently available evidence in a couple of minutes. Here's what the summary says:
Alternative methods are commonly used in patients with dermatologic diseases, with homeopathy being one of the most common. Homeopathy was developed by Samuel Hahnemann (1755–1843) and is based on the law of similars and the law of infinitesimals. It is a regulatory therapy where high dilutions of particular compounds are thought to induce a counterreaction in the organism. In dermatology, homeopathy is often used in atopic dermatitis, other forms of eczema, psoriasis, and many other conditions. To date, however, there is no convincing evidence for a therapeutic effect. There are only a few controlled trials, most of them with negative results. The few studies with positive results have not been reproduced. Acceptance by the patient seems largely based on counseling and emotional care rather than on objective responses to the homeopathic drugs.Not particularly convincing, is it?
Beech is at least unlikely to do himself any harm by trying homeopathy. However, real harm can result when people use homeopathy and avoid real medicine, as this tragic case in Australia shows. Nine-month old Gloria Thomas died after her eczema allowed an infection to get out of control. Her parents had been treating her with homeopathy instead of real medicine: they were later convicted of manslaughter by gross criminal negligence.
But the main reason that I have a bad feeling about this, is that even if Beech gets better, it won't prove anything about homeopathy (and the same goes if he doesn't see any improvement, to be fair). This is essentially an uncontrolled case report, and there would be no way of showing that any improvement resulted from the homeopathy, rather than, say, the natural cyclicity of eczema, or the placebo effect. Whatever happens to Beech, it isn't going to trump the evidence from the most recent systematic review of the evidence from clinical trials. So Beech's experiment cannot add anything to what we laughingly call the "debate" over the efficacy of homeopathy.
Still, good luck to him. I hope he does get some relief from his symptoms, but if he does the chances of it having anything to do with homeopathy are vanishingly small.