He doesn't want to pay it back and then be called in a year with the same threats because it wasn't initially documented.

The cancelled check and the email requesting it will suffice to prove you paid what was requested. Hang onto both, and snap a picture of them side by side should anyone come at you a year from now requesting the same amount again.

Now, we are assuming the math is correct. If it isn't, or if there is some dispute about how much exactly should be repaid, well that's a different question that might be a little less cut and dry. If the ex-employer asks for too little now, realizes the math error, and asks you for the difference on it a month or even a year later, they are fully in their legal right to do that and could take your family member to court if you refuse...as long as it isn't 15 years from now when the statute of limitations for claims relating to a written contract have long since expired.

To stave off risk of the above, as well as to protect yourself from a math error that leaves you repaying too much, it is worth double-checking the calculations. They are under no obligation to explain their math to you.

If you overpay, and you figure it out later, you too are entitled to ask the company to pay you back the over-repayment. Better for both parties to just make sure the math is right up front and handle it in one swoop though. As /u/Sorthum said, the tax situation is definitely going to be screwy enough already. It is probably worth paying a CPA for an hour of their time to discuss things now so you are prepared come April.