How should I deal with unlawful money I gave to a member of my family, given that I have repented to Allah, I cannot repay the full amount, and I cannot tell him the truth about where the money came from?

If the money was acquired unlawfully, such as through Riba (usury) or gambling, then one must repent and dispose of it by spending it on the welfare of Muslims or giving it to the needy. Your giving it to your needy relative absolves you of responsibility, and you are not obligated to inform him of its unlawful nature. However, if the money was obtained through theft or usurpation, it must be returned to its rightful owners, based on the hadith (definition): "The hand is responsible for what it took until it returns it." Then, you should compensate your relative if he has no right to it. Charity alone is not sufficient without returning the money to its owners.