Is it permissible to count the amount I lent to my deceased friend, who had many debts, as part of my Zakat, and to waive the claim for the debt from his family?

It is not permissible to waive a debt owed by a debtor and count it as zakat (definition), because zakat is a right of Allah and is disbursed to its specific recipients as outlined in the Quran. Waiving a debt and counting it as zakat is considered a collection of the money. In this situation, one of two things is preferable: either to give the debt to the debtor as charity, especially if he is in hardship, or to share proportionally with the other creditors from his assets if he left any.