Family Sponsor Immigration Act of 2001 - Amends the Immigration and Nationality Act to permit the acceptance of a signed affidavit of support (to establish that an alien is not excludable as a public charge) from another eligible family sponsor of the alien upon the death of the original sponsor-petitioner (after acceptance of the petition but before signing the affidavit of support) if the Attorney General determines for humanitarian reasons that the original sponsor's classification petition should not be revoked.