I have had success with essential oils in the past, but with staph we tried essential oils in the bath and topically on the sores, and had no success. We used repeatedly, for two months: tea tree, lavender, geranium, oregano oil, and the thieves blend. They didn't change the look of the staph sores one bit. We did not, however, run a diffuser.
We also did not try iodine.
We also washed everything religiously in Borax, every day. We also did cell salts, regular homeopathy, probiotics, epsom salts baths, colloidal silver, and calendula wash.
Honestly, after struggling with this thing for two months, my sense of it is that it is not so much a bacteria issue as an immune system issue. It is tempting to want to eradicate the buggers, but really I think the issue is balance with other bacteria -- staph is *supposed* to be on our skin; it's normal. Certainly the more aggressive kind is not normal, but I still think the "eradication" seems to work best from the inside out than from the outside in. So, we tried all sorts of powerful "germ killing" approaches, some of which were successful temporarily (the manuka honey being the most wildly successful) but I think the inner alchemy is what has finally brought it under control/in balance.
To further that example, I have never broken out in staph -- even though I sleep with him, touch the sores with my fingers, etc etc etc. Nor has anyone in our house, including our six year old, who never washes his hands. In my opinion, it is more than a simple bacteria issue.
Here's the link to the water that helped:http://www.mrsa30day.com/FAQ.shtml
Like I said, it sounds like a pitch (I was very suspicious at first), but it seems to work.
We use Pharmax probiotics, using the link in JaneS signature (she posted above.) We alternate that with Klaire infant probiotics.
Good luck! I know it sucks, and is stressful. Hopefully something in this thread will help your little one.