You could start by explaining how the internet was initially created by academic and government institutions for research purposes, and now BOINC is just another layer that gives researchers that ability to leverage the computing power of a much wider group of machines. Back to the basics?

It's a screensaver that runs while you aren't using your computer and works in the background to do things like search for signals from aliens and find a cure for cancer.

Hardly complete or 100% accurate but for a simple description for the average person to understand it's a pretty good start. If they ask questions you can provide more details.

Wow. Okay. I posted this question on several forums and have admittedly received a lot more response than I expected. There’s no way I can reply to everything so I’m just going to absorb and reply in general. 

I’m providing the links to all the threads (ordered from most responses to least). I would highly recommend reading them. There are very insightful responses that approach the question from several different perspectives. I have no doubt that they will help anyone looking to spread the Word of BOINC. I plan on integrating a lot of the ideas into my personal elevator pitch.

https://www.reddit.com/r/gridcoin/comments/d2oi12/what_is_boinc_i_need_help_explaining_to_my/
https://www.reddit.com/r/BOINC/comments/d2ohll/what_is_boinc_i_need_help_explaining_to_my/
https://setiathome.berkeley.edu/forum_thread.php?id=84643
https://steemit.com/gridcoin/@jringo/what-is-boinc-i-need-help-explaining-to-my-friends-family
https://www.worldcommunitygrid.org/forums/wcg/viewthread_thread,41828
https://boinc.bakerlab.org/rosetta/forum_thread.php?id=13277#91104
https://boinc.berkeley.edu/forum_thread.php?id=13106

My own answer to the question was lightly touched on by Mod.Sense over on the Rosetta@home forums when they said:

“BOINC is also the client which makes it easy for participants to connect to projects…”

Lumendan from the Gridcoin community on steemit also shared something that resonates with what I’ve been thinking:

“There is a network of people who contribute to research by volunteering some of their PC's spare computing power to solve massively difficult problems. BOINC is the backbone of that network, it's the system that links research projects with volunteers.”

To elaborate:

BOINC is the middleware software, but it feels like there’s more... particularly after having experienced the BOINC workshop... after sitting in a room (or walking along a lake) and discussing the worlds of BOINC and science with only a few dozen of the engineers, scientists, researchers, professors, contributors, builders, leaders, crunchers, thinkers, tinkerers, students, publishers, enthusiasts,... whatever, involved with BOINC... 

it feels like there’s more to BOINC than the code that handles the data.

It feels like there’s more in that when I explain BOINC to my friends we never end up discussing the software or how it works. And when I force them to start crunching and we start talking about BOINC again, we almost never talk about a feature or a coming update. And when an update is released my friends really don’t care. 

But when DHEP was spun up… or when DHEP was spun down… or as BlackHoles@home gets closer to release or when WCG teases new climate change projects… or when they realize that IBM runs WCG, or that DHEP was hosted out of the University of Sussex… or that LHC runs data with BOINC… or that some projects are hosted on laptops in closets by people just like them…

What I’m getting at, I suppose, is that these folks really couldn’t care less about the tech behind BOINC, supercomputers, or even science and research in general. Their interest seems to focus on the human network behind BOINC and the potential to become part of that network.

First they seem to think:

Here’s this human network established by and around a program that seeks to better humanity through the advancement of science, understanding, and knowledge. 

Then they seem to think:

Here’s this program developed, maintained, and utilized by this human network that lets me immediately participate, contribute, and interact with scientists, entities, and other like-minded individuals by completing real world scientific and data-analytic tasks. 

So what is BOINC? Well. What is the internet, what is Linux, what is Bitcoin?

Maybe BOINC is a human network anchored by a grid computing technology that enables seamless and immediate participation in and contribution to that network.

One thing I’ve learned for certain over my years interacting with the BOINC community is that BOINC feels like far more than just some software.

Middleware is a great answer if you are talking to a software engineer but its like trying to explain "what is light" or "what is time". If your audience doesn't have previous exposure to the technical background details it may be better to describe what is it's effect or how is it useful. 
My angle would probably be:

*There is a network of people who contribute to research by volunteering some of their PC's spare computing power to solve massively difficult problems. BOINC is the backbone of that network, it's the system that links research projects with volunteers.*
(You can exchange PC with phone, tablet, laptop or any other applicable device) 

If that raises interest I might explain how a big problem is broken up into little parcels of work and sent out to millions of volunteers, also, when the results are combined the network of volunteers can outperform a supercomputer.

Imagine an anthill, if you are the size of an ant, the hill and the nest that made it is like a city, pretty big!  There is a queen ant that co-ordinates all the workers, but they all also act independently, by collective effort large amounts of earth can be moved and structures built, by tiny ants.

BOINC allows someone to create a project like the queen in the ant-hill does, this co-ordinator breaks a big computational problem into ant sized parts, and sends them out to individuals to be completed and returned.  With enough ants, its amazing how much research can be completed.  Because the ants volunteer their computing power, it makes BOINC a very cheap for scientific problems, that could not normally be funded.

BOINC is a screensaver like software on your computer which helps scientists compute problems while you're not using your computer, and in return you can earn a crypto currency call Gridcoin.