First, a question: Does your mother have homeowner's insurance on the home? Whether she should file a claim depends on a few things, but she needs to decide quickly because if she waits very long to report what happened to her insurance company, it may refuse coverage. She should read her policy ASAP. I think she should probably file a claim and let the insurance company worry about suing the neighbor's friend.
Now the roof is leaking and my mother's tenant who lives ont he top floor (of two floors) has lots of water damage.
Your mother is probably not financially responsible for the damage to the tenant's belongings. I'm not a lawyer so I could be wrong, but I believe her having hired an unlicensed individual to repair her roof is not sufficient by itself to show she was negligent.
And we are sure he did a poor job which is causing this mess.
Your feeling sure of that does not count. Do you have evidence that is likely to convince a judge that the repair job was negligent and that the water damage resulted from that negligence? Exactly what did this man do wrong?
She's definitely learned her lesson to not do business worth that much without a trail.
Your mother's having paid in cash does not automatically mean she won't be able to convince the judge that this man did the work. Did she tell the neighbor that she had hired his/her friend to fix her roof? Even if that neighbor has since died or moved to Timbuktu, the roof guy would have to be an awfully good liar to pull off the lie that he never worked on her roof.
Your mother may want to make an appointment with a lawyer in her state (the state where this home is located) for an opinion on both her possible negligence and that of the neighbor's friend. But first I think she should make sure she knows exactly what happened, so in addition to getting her estimates for fixing the roof she should also ask for the contractors' opinions on what caused the leak.
One final word of caution: it is very possible that this man has spent her money and does not have enough attachable assets (or garnishable wages) to pay her even if she wins a judgment against him. So if she can sue in small claims court without hiring a lawyer, that would probably be wise.