I presume you are asking whether there is any merit to such a lawsuit. This will depend on whether or not prison officials committed any negligence or misconduct. The prison is responsible to act within a standard of care, which includes duties to: take reasonable safety precautions, to supervise inmates, and to treat prisoners in accordance with federal regulations, constitutional and legal standards, and accepted human rights standards. Given that some inmates are extremely violent or mentally unstable, doing all of this does not afford a 100% guarantee of safety. Whether or not the prison will have liability for the injuries will depend on whether or not the prison personnel adhered to these requirements or not.
Your family should consult an experienced civil rights or prisoner's rights attorney. You should also ask the watch commander of the facility if they have investigated the circumstances of the injury, and if there is a report or other information available, and any inmates or others who may be responsible are being prosecuted. These documents might give an experienced attorney some insight as to whether a suit has merit.
Another important issue is whether or not your cousin remains in danger of further attacks or retaliation. If this is the case, he should ask his counselor or watch commander about protective custody, or transfer to another unit or facility.
Yes -- maybe.
The FTCA and §1983 actions can be brought against the government. Your cousin's family should contact an attorney (or two) to see if there is something that can be done for him.
Not all personal injury lawyers will take prisoner cases, they may start with his criminal attorney.
Just because he is a prisoner does not mean he loses all of his rights and protections!
There are many attorneys that would handle a case like this. The only issue would be trying to show how the prison failed to live up to their responsibilities. I wish you the best and certainly discuss this with an attorney that specializes in these actions.