I recently implemented a Wiki implementation (XWiki Enterprise) for a company to collaborate on building a knowledgebase for employees.
It has document attachment and conversion (imported docs into wiki format), auto-linking of related pages, detailed revision history, permission sets, etc, as well as automated notifications for updated pages, discussion pages and forums, etc. You can even track projects and tasks in it. It also has built-in media handling plugins (so you could host those video interviews you mentioned and play them in the browser from the wiki).
Such a thing would be awesome for big investigations like this, so the huge teams of investigators would collaboratively build a wiki for the investigation. For big cases like the one you mentioned (and the Trump investigation) I would devote an entire instance just to that one investigation.
The intelligence community has their own internal wiki called Intellipedia (three wikis actually, one for each level of clearance).
No one seems to have mentioned the best one out there: xwiki. http://www.xwiki.org/xwiki/bin/view/Main/WebHome
Its better than confluence, more free, and better setup. Integrates with AD etc, has all the wisiwyg stuff you need. Indexes attachments, allows uploading of office docs to convert to pages as well.
You should check out XWiki. I installed it back at version 8 and it was a pain but the newest version was a breeze to install. It has great extensions, supports a variety of Syntaxes, has been around a surprisingly long time.
http://www.xwiki.org/xwiki/bin/view/Documentation/AdminGuide/Installation/InstallationViaAPT/
Sharepoint is just a web framework that can be used in different ways. Possibly a closely comparable open-source solution is XWiki, in Java. But what you do with it drives the tooling.
Originally used Microsoft OneNote, as others have recommended. I now move many things to a self hosted XWiki. I considered a few other wiki engines, but went with XWiki because:
Not sure your experience level on setting up home servers or but it's going to be very similar to that, maybe stand up an internal tomcat and install xWiki on that and boom your almost done, just google any problems you come across and xwiki doc is pretty good for admin stuff on that end of things. The hard part will just be figuring out the IT infrastructure for the company. Figure out if an 'intranet' is even what they really need and if they would be better off just hosting an xwiki on a private host with an ip white list for the company. There are a lot of routes you could take here.