Amoxy is not very powerful against lyme, it certainly does not work without other
antibiotics taken together. The usualy treatmetn for
kids under 8 is cefuroxim twice daily (high dose proprtional to body weight) plus azithromycin 3 times a week.
So long as you act within 6 weeks of the bite he should be fine. Once you are past that stage, it could be harder to cure and I personally would raise heaven and earth to get my son on the right antibiotics before it gets to that.