Hi H.,
My 6yo son just had the same thing happen. The day after we returned from a trip to NH, I noticed a bullseye shaped rash surrounding a bug bite on his arm. But it looked like a rash, not a bruise. (It was red, not blue. And slightly raised/bumpy.) I took him to our doctor's office. They said it looked like text-book lyme disease and that the protocol was to go ahead and treat with antibiotics rather than testing first. (Because the test for lyme is not 100% accurate.) They said that one course of antibiotics would totally treat it since we caught it early.
Have you been traveling back east? It's my understanding that lyme disease is carried by deer ticks, which are common in the northeast but are not found out here (CA).
They gave my son a three times per day for 14 day course of antibiotics. This is a higher dosage than the normal twice a day for 10 days dosage that they give for things like ear infections (which he had just had the month before).
In my son's case, the rash was gone within 24 hours after starting the antibiotic. If your son still has the rash/bruise, perhaps you should take him for a second opinion. I'm not sure they gave you a strong enough dose of antibiotic. But if the rash is already gone, then there is probably not much point of going. It might be worth a phone call or two, but unless you have documentation from the first doctor that the diagnosis was lyme disease, I think it would be difficult to get another doctor to give you a new prescription.
I hope this helps,
D.