Describing that we (as systems' owners) can leverage users' capabilities to do something useful unpaid, what's the best word/expression for?
I'm looking for something rather than Crowdsourcing.
If your target is something more than just content how about a
The most appropriate answer that I just found is Human-based computation (also Human computation):
Human-based computation (HBC), human-assisted computation,ubiquitous human computing or distributed thinking (by analogy to distributed computing) is a computer science technique in which a machine performs its function by outsourcing certain steps to humans, usually as microwork. This approach uses differences in abilities and alternative costs between humans and computer agents to achieve symbiotic human-computer interaction. (Wikipedia)
And another evidence from Luis von Ahn, a well-known scientist from CMU:
I build systems that combine humans and computers to solve large-scale problems that neither can solve alone. I call this Human Computation, but others sometimes call it Crowdsourcing. Some of my past projects include: CAPTCHA, The ESP Game (acquired by Google), GWAP, and reCAPTCHA (acquired by Google).
Last Edit : Collective Intelligence is another synonym.
Mind meld, group think, 42 heads are better than one, human cloud (or cloud of humanity), hive mind.
It sounds to me you want to foster volunteerism on some kind of social networking platform. So, you might want to call it something like a Internet community service platform.