You can try, but if cousin is insolvent, you may only obtain a worthless judgment. Your new contractor will not do the work under these circumstances. You will need to pay new contractor, then seek reimbursement from your cousin. Good luck!
Your question doesn't provide enough facts. Did you pay your cousin in full, or only partly? Either way, your "damages" for his not finishing are the total costs paid to both contractors minus what he agreed to do the work for; that's the "cost overrun." As noted by Ms. Smith, you can sue your cousin to get money back, but ability to collect is the open question. If nothing else, however, this will give you a basis to refuse to hire your cousin ever again!