Homeopathy: Activates the body's own natural healing processes.

Short answer:

• No it doesn't - there is no biologically plausible method for this to occur, and there is no credible evidence for it having ever occurred.

A little more:
This little number is often explained as operating a little like a vaccine; e.g. the remedy somehow tells the immune system "look for things that cause symptoms like this" and that stirs it into action. Of course this a completely false analogy, and it wouldn't work even if the analogy were valid.
Vaccines work by introducing a harmless form of an actual microbe into the body. The immune system produces antibodies against that microbe and this enables it to be prepared for invasion of the genuine microbe (or ones very like it) should it encounter it at a later date.
Homeopathic remedies with "potencies" beyond 12c do not contain anything other than water, and even below that the content is negligible. There is nothing present for the immune system to learn from. Even if there were still some of the original preparation present, this would not stir the immune system into life when encountering a genuine illness-inducing microbe. Most of the the "mother tinctures" for those homeopathic remedies that are used for microbial diseases bear no structural relationship to the actual cause of the malady, and thus immune system will not recognise the real thing when it arrives. Additionally remedies are rarely prescribed as preventatives (with the exception of "nosodes" like the one idiotically prescribed for malarial prophylaxis), there are usually given when the disease is already present and the immune system is already fighting the disease, and has no need of such activation.
Add to this the fact that homeopathic remedies are also prescribed for disorders where the immune system has little or no involvement (like anxiety or type-2 diabetes), and for auto-immune related disorders (coeliac disease, rheumatoid arthritis) where the problem is that the immune system perceives part of the sufferer's own body as an enemy, and all that is left is a supernatural explanation for homeopathy's supposed mechanism of action. Some would attempt to counter these arguments by saying the immune system thing is just an illustrative analogy. They are then hard pressed to described any plausible actual mechanism by which their claims may be substantiated. Homeopaths' talk of vital forces and miasmas is just medieval thinking, and this whole argument of activating the body's own natural defences is simply superficially plausible nonsense.