Sorry to hear your son is having such a hard time FF.
I've seen a homeopath regularly with my DD (now two) and find it helpful. I am aware of the views / studies that it is nothing but a placebo. however, there are also views / studies that show the power of the placebo (there's a recent BBC documentary looking at the Harvard research medics studying the placebo effect).
I also think that there is a difference between good research evidence which proves something does not work, areas where there is a lack of good research evidence (see recent Jenette Winterston article on fasting www.theguardian.com/books/2015/jul/11/jeanette-winterson-why-i-fasted-11-days quote from Dr Andreas Michalson 'If I had been studying a new drug and got these results, I would be getting phone calls every day. It is very easy for critics to say there are not enough studies when we know there is no funding for these studies.'
There is plenty of funding for drug studies, obviously.
For me, I am open minded on whether homeopathy 'works' directly. I certainly think there are many things in our work that cannot, or we have not yet understood how to measure and quantify. My personal experience is that I have found it helpful, but mostly to address combined emotional / stress and physical situations.
More important, my sessions with her give me a chance to reflect on our lives and explore and understand what needs to change to make things better.
For example, I have gained understanding of and confidence to trust our bodies own natural immune responses. DD, now two, has only had antibiotics once in her life, despite being prone to chest infections. Yes, her temperature goes high, but I stay with her, feed her, support her while that temperature kills the infection (that's its purpose), it comes down quickly on its own. 'Controlling' a temperature with paracetamol just makes it harder for the body's immune system to fight infection on its own. I always consult a doctor early to check it is a minor / normal infection (ie not meningitis) and supervise her closely.
She had recurring thrush, which I treated the first time with the medical / big pharma offer (canesten) but of course it returned. I worked out through research how to support my DD's body to deal with it herself (grapefruit seed extract spray - I make my own, and coconut oil with 10% tea tree and lavender oil instead of commercial nappy creams). It has never come back (over a year now) and she is nappy rash free.
Personally, I started learning about alternative approaches when I had debilitating IBS in my early twenties, with related acne. I now have a very happy digestive system through homeopathy, accupuncutre, diet and lifestyle changes. My skin is also great.
It sounds like you have given the medical approach for your son a serious go. Two years is a long time. Maybe it is time to stand back and take a more holistic view. As I understand it, the kind of eczema you describe is an expression of crisis in the immune system. Antibiotics, and multiple courses of antibiotics will certainly be driving that. have a look at this article, also from the Guardian www.theguardian.com/lifeandstyle/2015/may/18/truth-about-poo-doing-it-wrong-giulia-enders-squatting about the work of a medical macrobiologist 'Our bacteria fight pathogens, are involved in blood-group development, digest our food, extract energy, produce hormones and can affect our mood. This gut/brain connection is a fairly new area of medicine, which Enders is very excited about'.
Homeopathy might be the start of a different approach for you and your son. You also might want to talk to a nutritionist. I would guess that a good (really good, not just a yakult thing) probiotic might well help your son's gut flora start to recover.
I've also found acupuncture to be incredibly powerful (which I think, has been 'proved' to be effective). Any of these approaches are a good place to start. Someone good will likely make further suggestions after they've taken a comprehensive look at what is going on for your son, and you.
Good luck