What is the Shar'i ruling on a sister's refusal to put in writing that which proves her brother's right to the money he lent her to pay off her debt, and what are the verses and Hadiths that support the brother's position and might soften his sister's heart to return the right?

Documenting contracts and debts is commendable in Islamic law, as it preserves rights and prevents disputes. This is supported by the Almighty's saying: "O you who have believed, when you contract a debt for a specified term, write it down." You have the right to ask your sister to document the debt, and she should not refuse, for the appointed times are in Allah's hands. Whoever takes people's money with the intention of repaying it, Allah will repay on his behalf. If she is wealthy and capable of repayment, then it is forbidden for her to procrastinate.