Cisco Solution (Score 1)

So this solution is not open source or in the "cheap" range, but it will be pretty bullet proof with full support from Cisco

Look these pieces running together(Solution 1):

-Cisco Jabber (for use with Jabber guest which lets you connect via website to a video enpoint)
-Cisco Callmanager with VCS Expressway (your brains, and it lets physical endpoints an Jabber clients connect over the internet)
-Cisco TMS (Telepresence manager) , this lets you setup the endpoints to auto connect a video call at a scheduled time.
-Cisco DX 80s or 70s, there are not horribly expensive and they are full video endpoints that run android

This lets you have a video conference start a scheduled time and the outside party(with reference to your organization) can connect using jabber guest. Also, alot of the backend pieces are not available as cloud services which reduces your admin effort day to day.

Solution 2:

I would look at the blue jeans conference service. They support a plethora of units (sip and h323) to connect to a conference so you could use something like a polycom VVX or yealink (although the DX 70 price already gets near the VVX price).

If you dont have a team to back you up I would stay away from the Asterisk / SipXecs /Jitsi solutions. They will need engineer/s to manage and develop the platform. With those solutions you usually trade off freedom/free/customizability for ease-of-use (so you can make a really powerful and customized solution but you will need an engineering team to make it all work together).