The "Exclude From Enterprise Lists" option is located in Site Setup > Site > Enterprise > Exclude From Enterprise Lists. When an Initiative is selected with the option "Exclude From Enterprise Lists", End users cannot search for ideas from that Initiative in Enterprise.
Ideas or comments within Initiatives that have this option enabled would not trickle up to the Enterprise Updates feed or the Enterprise profile page.
Keep in mind that the information that is hidden will only affect End-Users. Administrators will see all activity regardless.