My first association for the most (whatever that means) important invention was the unconscious mind, because, I thought to myself, the concept offers some explanation — and thus hopefully later remedies — for the behaviors coming from the darker sides of our nature.
Armed with better understanding of the origins of such behavior, hopefully we could fashion ways out of the irrational clamp that fundamentalist religion, blind nationalism or deep ideology often puts on our conscious awareness. But, one thought later was that I believe the unconscious does indeed exist, so logically it is a discovery, not an invention.
That (somewhat uneasily) suggests psychotherapy (again, whatever that means given all its incarnations — psychotherapy being but one of a basket of techniques to make the unconscious more conscious) as my invention. At least in principle — and often in practice too I believe — it does offer us the opportunity to become more conscious, therefore less inclined to absolute thinking and the subjugation and/or violence absolutism often engenders in the minds of those who don't harbor doubts.
In discussing the causes and possible solutions to global environmental problems (e.g., global warming in particular), I note in dialogues with junior high school students — right on down to senate committees — that we can't easily fix problems we can't see. Thus, solutions to long-term, global-scale systems threats require — in a democracy at least — overcoming any collective denial that our "puny" individual impacts can cause a major disruption at a planetary scale or over timeframes longer than our lifetime.
Admittedly, I'm not going to seriously claim psychoanalysis is as "important" an invention as the scientific method over the past 2000 years (as I recall one of your respondents proposed). What I see as a key invention for the year 2000+, though, is an expanded systems analysis that includes methods to build in an understanding of the role of the unconscious of individuals which leads to lifestyles and behaviors which "scale up" to create unanticipated collective consequences.
Although not directly responsive to your question, the invention I really like — think we will really need — is a fusion of systems analysis with psychotherapy. But the new field of "systems therapy" is yet to be invented!, leaving me dangling uneasily between systems — and psycho-analysis. Perhaps, if armed with insights from tools that integrated physical, biological, social and psychological drivers of our behaviors across a range of scales, rather than always chugging merrily along in business as usual mode, we'd be more aware of the range of consequences of our unconsciousness. Then, if we continued to damage the collective or put the future at risk, at least that would be more of a choice and less of a surprise. With best wishes to all for the holiday season.