How to add "Family Member Information" Columns to Event Attendee List report?
All the checkboxes in the screenshot you posted refer to built-in CiviCRM fields. "Family Member Information" doesn't refer to any built-in CiviCRM field, so it's a bit hard to know exactly what you're asking to do.  Here are some options:


If "Family Member Information" is the name of a custom field group, then you can add those checkboxes to the report by going to Administer menu » Customize Data and Screens » Custom Fields.  Find the Family Member Information custom group and click View and Edit Custom Fields.  Change any field you'd like to appear on this report such that Searchable is set to Yes.
If "Family Member Information" is set via a relationship, you have a few options.  I won't detail them here unless you update the question to indicate this is the case, but:


Add folks to a group, use the Relationship Report.
The Extended Reports extension has an Event Attendee List report with relationships.
If you're using Drupal, this is easy to implement using Views.