"Counterculture" is such a catch all term, and often the Civil Rights Movement is not included in its definition, but it's hard to argue that anything was much more counter to established culture, or much more successful at achieving its goals than that movement.
I'd also give a shout out to Students for a Democratic Society, which influenced the anti-war movement, education policy and offerings, and gave, at least initially, a pretty focused direction for youth to pursue change in.
I would agree that the "hippie" movement was probably the most influential movement of the 60's. They were involved in the anti-war protests against the Vietnam War, they also were responsible for many changes in our culture during that time.
Sex, drugs, rock'n'roll, free love. The hippie movement is clearly one that had a profound effect on Western society throughout the sixties in terms of its focus on opposing American foreign policy at the time and challenging a number of social values of the day. I think there is still a little bit of a hippie left in all of us, however small, and the way that it came to represent a anti-establishment movement makes it memorable.
I think the anti-war movement was the most influential "counterculture" of the 60's. Policies in Vietnam were absolutely influenced by the demonstrations and voices of the young people back home calling for an end to war. More than that though, our attitude towards soldiers and those in the military has changed because of this movement. No one would have had anything but respect and celebration for returning soldiers after the World Wars. Military service was respected and considered a patriotic duty. During the war in Vietnam, veterans came home to no fanfare and no homefront, just anger directed at them for doing their job. The reality is that much of America still responds this way. We don't have the respect for our service men and women that existed before Vietnam. Before the antiwar movement of the 60's, you supported the war effort simply because you supported the country (even if you disagreed, as many did). Now we see a lack of respect for our military and almost no true homefront for our soldiers on the front lines to return to.
Although not an organized social organization, the hippie lifestyle was certainly one of the most popular in America during the late 1960s and early 1970s. Millions of young Americans adopted aspects of hippiedom--long hair, drugs, wildly colored clothes, free sex, and militant political stances. Many outgrew the lifestyle, but hippies remain today. Many have grown old and conformed to standards held by their parents, but many have passed down their beliefs to newer generations.
The most influential element of the counterculture was the movement by the baby boomer generation (who were in college at the time) to embrace peace as a cause, in opposition to the war in Vietnam. Most were alienated by the war, racism, and the cynical politics of the day. They did not adopt the violence of the new left, but rather turned to drugs (primarily LSD), rock music, long hair, sexual freedom, and rock music. Young men wore their hair long and grew beards, and young women went bra less. They called themselves the "beautiful people" but were known by others as "hippies" and "flower children." The motto of the times, as expressed by Timothy Leary was "tune in, turn on, drop out." The Counterculture was largely responsible for the success of the Woodstock Music Festival.
Many members of the counter culture followed Oriental mystical religions, such as Bahai; and were keen followers of transcendental meditation as taught by the Maharishi Mahesh Yogi of India. The counterculture added new phraseology to the English language, such as "bread" for money, and "threads" for clothes.
Peace, love, drugs and sex did not make for a tenable lifestyle, however. One critic said the Counterculture made "more babies than bread." Although many attempted life in communes, they were ill prepared for life in nature, and most communes quickly failed. The former hippies and flower children are today stock brokers, lawyers, and other professionals. In short, the hippies became the yuppies.
I assume that you are thinking about various movements in the '60s as separate countercultures. In other words, I assume that you are saying that the Women's Lib people were one counterculture, the Hippies another, and so on. If so, I would argue that the Women's Lib/Feminist counterculture was the most important and influential.
I would argue this because I believe that the women's movement did the most to change US society. I think that many of their goals were achieved to a greater degree than those of other countercultures. I also think that the movement had impacts that would have been difficult to foresee at the time.
Just to give one example, I am male but have had only part-time jobs since my first child was born. At that point, I quite my full-time teaching job to stay home with the child (later, the children) while my wife worked. I know quite a few other men who do this. That would have been unheard of in my parents' generation (they were adults before the counterculture days).
As another example, my mother was not allowed to wear pants on the campus of her public university in the early 1960s. That sort of paternalistic attitude towards women has changed drastically.
Finally, the feminist counterculture has had other impacts that might not have been anticipated. For example, the American family has changed greatly since the '60s. Divorce and remarriage are common now. This is something that has come about as women have gotten more rights, more education, and more of the mindset that they should assert themselves in relationships and demand fulfillment.
So, I would argue that the feminist counterculture was the most influential of the era.