this is NOT legal advice, this is just my layperson's "gut feeling", but you cant sue someone for not having insurance when the only damage arising out of an incident was to that person's uninsured property. also, id bet your buddy has a cause of action in tort (negligence) against the kid who crashed it (but he should call a lawyer to check that out).
the only thing is, your buddy presumably gave the kid permission to ride the bike, and that might mean, depending on which theory of law ohio follows, that your buddy assumed the risk that something like that would happen, which might bar any sort of recovery against the crasher. annnnd, although the kid who crashed it probably cant sue your buddy for not having insurance, i'd bet that a court would still probably say that your buddy further assumed the risk by not insuring it, and might not be able to recover anything.
and as a long shot, your buddy might have some sort of contract claim, but it doesn't sound like he gave any consideration for the promise, since the promise to pay for the bike was made after the crash, which makes me think the contract claim is a long shot.
and finally, as im not an attorney, there might be a whole host of other claims, especially under specific ohio statutes, that i just have no clue about.
tell him to CALL A LAWYER. most of the time, a quick opinion over the phone as to whether he has a cause of action will be free, and whatever he pays for in attorneys fees might suck, but at least it will give him a shot of getting some money for the bike, whereas it seems pretty unlikely that hell see a dime if he doesnt get a lawyer involved.
all of that is assuming worst case scenario (where the thing goes to court), by the way. usually, if you get an attorney involved, the matter will get settled well before it gets in front of a judge. id be willing to bet that the other family would fork over some money as soon as they got a letter from a lawyer. maybe not the whole amount, but who knows. they could quickly spend waaaay more than the cost of the bike on legal fees by trying to fight it out in court, so instead, theyll probably be willing to compromise and make your buddy go away (especially when they talk to another lawyer, who tells them - assuming that there arent any other key facts to the incident that you left out - that they most likely have no chance at a counterclaim against your buddy). but like i said, im not a lawyer, and an attorney could probably really help him out, so tell him to call a lawyer.